Big Ideas Math Book 8th Grade Answer Key Chapter 7 Functions

Big Ideas Math Grade 8 Ch 7 Functions Solutions provided here cover the questions from Lessons 7.1 to 7.5, Practice Tests, Review Tests, Cumulative Practice, Assessment Tests, etc. You can have deeper insights into all the Topics of Functions such as the representation of functions, linear functions, analyzing and sketching graphs, etc. Use the below available quick links for Big Ideas Math 8th Grade Chapter 7 Functions Answer Key and clear all your ambiguities regarding the concerned topics.

STEAM Video/Performance Task

• STEAM Video/Performance Task

Getting Ready for Chapter 7

• Getting Ready for Chapter 7

Lesson 1 Relations and Functions

• Lesson 7.1 Relations and Functions
• Relations and Functions Homework & Practice 7.1

Lesson 2 Representations of Functions 

• Lesson 7.2 Representations of Functions
• Representations of Functions Homework & Practice 7.2

Lesson 3 Linear Functions

• Lesson 7.3 Linear Functions
• Linear Functions Homework & Practice 7.3 

Lesson 4 Comparing Linear and Non Linear Functions

• Lesson 7.4 Comparing Linear and Non Linear Functions
• Comparing Linear and Non Linear Functions Homework & Practice 7.4

Lesson 5 Analyzing and Sketching Graphs

• Lesson 7.5 Analyzing and Sketching Graphs
• Analyzing and Sketching Graphs Homework & Practice 7.5

Functions Connecting Concepts

• Functions Connecting Concepts
• Functions Chapter Review
• Functions Practice Test
• Functions Cumulative Practice

Functions STEAM Video/Performance Task

STEAM Video

Apparent Temperature
Sometimes it feels hotter or colder outside than the actual apparent temperature. How hot or cold it feels is called the temperature. What weather factors might contribute to the apparent temperature?
Watch the STEAM Video “Apparent Temperature.” Then answer the following questions.

1. Robert says that the Wet-Bulb Globe Temperature (WBGT)index is used as a measure of apparent temperature.

In the formula, T_W is the natural wet-bulb temperature, T_G is the black-globe temperature, T_D and is the dry-bulb temperature. Find WBGT when T_W = 75ºF, T_G = 100ºF, and T_D = 84ºF.
2. Different categories of Wet-Bulb Globe Temperatures are shown in the chart. Each category can be represented by a different-colored flag. Which flag color is displayed when WGBT = 87.5ºF?

Performance Task

Heat Index
After completing this chapter, you will be able to use the STEAM concepts you learned to answer the questions in the Video Performance Task. You will be given information about heat index.

You will be asked to create a graph of the temperatures and heat indices. Why is it useful to know the heat index?

Functions Getting Ready for Chapter 7

Chapter Exploration

Work with a partner. Copy and complete the diagram.

1.  Answer: ( 1 , 2 ) , ( 2 , 4 ) , ( 3 , 6 ) , ( 4 , 8 ) .

Explanation:
Given , Width w = 2 , Length l = x  where x = 1 , 2 , 3 , 4 .
To find Area A of a rectangle we have , A = w × l
for x = 1 , A = 2 × 1 = 2 ,
for x = 2 , A = 2 × 2 = 4 ,
for x = 3 , A = 2 × 3 = 6 ,
for x = 4 , A = 2 × 4 = 8 ,
So, for every value of Input x = 1 , 2 , 3 , 4 we have Output A = 2 , 4 , 6 , 8 , respectively .
That is ( 1 , 2 ) , ( 2 , 4 ) , ( 3 , 6 ) , ( 4 , 8 ) .

2. Answer: ( 1 , 6 ) , ( 2 , 8 ) , ( 3 , 10 ) , ( 4 , 12 ).

Explanation:
Given , Width w = 2 , Length l = x  where x = 1 , 2 , 3 , 4 .
To find Perimeter of a rectangle we have , P = 2( l + w )
for x = 1 ,P = 2( 1 + 2 ) = 2 × 3 = 6 ,
for x = 2 , P = 2( 2 + 2 ) = 2 × 4 = 8  ,
for x = 3 , P = 2( 3 + 2 ) = 2 × 5 = 10  ,
for x = 4 , P = 2( 4 + 2 ) = 2 × 6 = 12  ,
So, for every value of Input x = 1 , 2 , 3 , 4 we have Output P = 6 , 8 , 10 , 12 , respectively .
That is ( 1 , 6 ) , ( 2 , 8 ) , ( 3 , 10 ) , ( 4 , 12 ) .

3. Answer : ( 1 , 6 ) , ( 2 , 12 ) , ( 3 , 18 ) , ( 4 , 24 ) .

Explanation:
Given , Radius of a circle , where as  r = 1 , 2 , 3 , 4
To find the circumference of a circle , we have C = 2Òr , Ò = 3.14 , or we can write it as 3 .
for r = 1 , C = 2 × 3 × 1 = 6 ,
for r = 2 , C = 2 × 3 × 2 = 12 ,
for r = 3 , C = 2 × 3 × 3 = 18 ,
for r = 4 , C = 2 × 3 × 4 = 24 ,
So, for every value of Input r = 1 , 2 , 3 , 4 we have Output C = 6 , 12 , 18 , 24 , respectively .
That is ( 1 , 6 ) , ( 2 , 12 ) , ( 3 , 18 ) , ( 4 , 24 ) .

4. Answer: ( 1 , 9 ) , ( 2 , 18 ) , ( 3 , 27 ) , ( 4 , 36 )

Explanation:
Given , Two Edges of a cube = 3 , h = 1 , 2 , 3 , 4
To find the Volume of the cube we have , V = a³
for h = 1 , V = 3 × 3 × 1 = 9 ,
for h = 2 , V = 3 × 3 × 2 = 12 ,
for h = 3 , V = 3 × 3 × 3 = 27 ,
for h = 4 , V = 3 × 3 × 4 = 36 ,
So, for every value of Input h = 1 , 2 , 3 , 4 we have Output V = 9 , 18 , 27 , 36 , respectively .
That is ( 1 , 9 ) , ( 2 , 18 ) , ( 3 , 27 ) , ( 4 , 36 ) .

Vocabulary
The following vocabulary terms are defined in this chapter. Think about what each term might mean and record your thoughts.
input
mapping diagram
nonlinear function
output
linear function

Answer : Input : The ordered pairs can be used to show inputs and outputs of a function ,The input is the number you feed into the expression, and the output is what you get after the calculations are finished.

mapping diagram : A relation pairs inputs with outputs , A relation can be represented by ordered pairs or a mapping diagram .

nonlinear function : nonlinear functions are functions which are not linear. Quadratic functions are one type of nonlinear function. It is a relation between two variables , function that does not form a line when graphed.

output ; The ordered pairs can be used to show inputs and outputs of a function ,The input is the number you feed into the expression, and the output is what you get after the calculations are finished.

linear function : A linear function is a relation between two variables that produces a straight line when graphed. And it has one dependent variable and one independent variable .

Lesson 7.1 Relations and Functions

EXPLORATION 1

Interpreting Diagrams
Work with a partner. Describe the relationship between the inputs and outputs in each diagram. Then complete each diagram. Is there more than one possible answer? Explain your reasoning.

Answer: a. The relation between the inputs and outputs is outputs are the result of twice as many times the inputs.
b. The relation between the inputs and outputs is outputs are the result of colors of inputs . In this case we can notice that , for any one input we can have more than one output .

Explanation:
a. As shown in the diagrams , The relation between the inputs and outputs is outputs are the result of twice as many times the inputs , so for input 1 = 1 × 1 = 1 as output ,
for input 2 = 2 × 2 = 4  ,
for input 3 = 3 × 3 = 9  ,
for input 5 = 5 × 5 = 25 ,
for input 8 = 8 × 8 = 64 ,
for input 9 = 9 × 9 = 81 ,

So, for every value of Input = 1 , 2 , 3 , 5 , 8 , 9 , we have Output = 1 , 4 , 9 , 25 , 64 , 81 , respectively .
That is ( 1 , 1 ) , ( 2 , 4 ) , ( 3 , 9 ) , ( 5 , 25 ) , ( 8 , 64 ) , ( 9 , 81 ) .

b. The relation between the inputs and outputs is outputs are the result of colors of inputs . 
for input Blueberry = color is blue as output
for  input lemon = color is yellow as output
for input Apple = color is yellow , red and green as output
for input Grape = color is green as output.

In this case we can notice that , for any one input we can have more than one output .

EXPLORATION 2

Describing Relationships Between Quantities
Work with a partner. The diagrams show the numbers of tickets bought by customers for two different plays and the total costs (in dollars).

a. For each diagram, how many outputs does each input have?
b. Describe the prices of tickets for each play.
c. A person buys 4 tickets for each play. Can you determine the total cost of all 8 tickets? Explain.
Answer:
a. For Play A ,The number of inputs are equal to number of outputs ,
For Play B , The number of inputs are not equal to number of outputs .
So, input 1 = 2 outputs , input 2 = 3 outputs , input 3 = 4 outputs .

b. For Play A , The price of the each ticket is $8 .
For Play B , The price of each ticket is $4 or $8 .

c. For Play A , each ticket is $8 , Then for 4 tickets = 4 × $8 = $32 .
For Play B , each ticket is $4 or $8 , Then for 4 tickets = 4 × $8 = $32 . or 4 × $4 = $16 .

Explanation:
a. For Play A ,
The number of inputs are equal to number of outputs , 4 inputs = 4 outputs
That is ( 1 , 8 ) , ( 2 , 16 ) , ( 3 , 24 ) , ( 4 , 32 ) .
For Play B ,
The number of inputs are not equal to number of outputs , 3 inputs are not equal to 7 outputs
That is , for input 1 = 4 , 8 as outputs ,
for input 2 = 8 , 12 , 16 as outputs ,
for input 3 = 12 , 16 , 20 , 24 as outputs ,
So, input 1 = 2 outputs , input 2 = 3 outputs , input 3 = 4 outputs .

b. For Play A ,
The price of the each ticket is $8 .
For Play B ,
The price of each ticket is $4 or $8 .

c. Given , A person buys 4 tickets for each play.
For Play A , each ticket is $8 , Then for 4 tickets = 4 × $8 = $32 .
And for 8 tickets = 8 × $8 = $64 .
For Play B , each ticket is $4 or $8 , Then for 4 tickets = 4 × $8 = $32 . or 4 × $4 = $16 .
And for 8 tickets =8 × $4 =$32  or 8 × $8 = $64 .

Try It

List the ordered pairs shown in the mapping diagram.
Question 1.

Answer: Ordered pairs are ( 0 , 12 ) , ( 2 , 10 ) , ( 4 , 8 ) , ( 6 , 6 ) .

Explanation:
As shown , Ordered pairs are the combinations of input and output
So , Ordered pairs are ( 0 , 12 ) , ( 2 , 10 ) , ( 4 , 8 ) , ( 6 , 6 ) .

Question 2.

Answer: Ordered pairs are ( 1 , -1 ) , ( 1 , -2 ) , ( 2 , -3 ) , ( 2 , -4 ) .

Explanation:
As shown , Ordered pairs are the combinations of input and output
So , Ordered pairs are ( 1 , -1 ) , ( 1 , -2 ) , ( 2 , -3 ) , ( 2 , -4 ) .

Determine whether the relation is a function.
Question 3.

Answer: The relation is not a function

Explanation:
The each input has  more than two outputs , Even one of those inputs are unclear of outputs
So , The relation is not a function .

Question 4.

Answer: The relation is a function .

Explanation:
Each input has exactly one output ,
So , The relation is a function .

Self-Assessment for Concepts & Skills
Solve each exercise. Then rate your understanding of the success criteria in your journal.

Question 5.
PRECISION
Describe how relations and functions are different.
Answer: Relations are nothing but the ordered pairs with Inputs and Outputs . On the other hand , Functions are The relation that pairs with one input with exactly one output  are called Functions.

IDENTIFYING FUNCTIONS List the ordered pairs shown in the mapping diagram. Then determine whether the relation is a function.
Question 6.

Answer: The ordered pairs are ( 10 , 1 ) , ( 15 , 1 ) , ( 20 , 13 ) , ( 25 , 7 ) and The relation is a function .

Explanation:
As shown , The ordered pairs are ( 10 , 1 ) , ( 15 , 1 ) , ( 20 , 13 ) , ( 25 , 7 ) .
Each input has exactly one output ,
So, The relation is a function .

Question 7.

Answer: The ordered pairs are ( 0 , -5 ) , ( 0 , -4 ) , ( 1 , -4 ) , ( 2 , -3 ) , ( 3 , -2 ) and relation is not a function .

Explanation:
As shown , The ordered pairs are ( 0 , -5 ) , ( 0 , -4 ) , ( 1 , -4 ) , ( 2 , -3 ) , ( 3 , -2 ) .
The input 0 has more than one output ,
So, The relation is not a function .

Question 8.
OPEN-ENDED
Copy and complete the mapping diagram at the left to represent a relation that is a function. Then describe how you can not modify the mapping diagram so that the relation is a function.

Answer: ordered pairs are ( -8 , -4 ) , ( 0 , -2 ) , ( 8 , 0 ) , ( 16 , 2 ) . To have the relation as a function we must have only one output for one input.

Explanation:
The ordered pairs of the diagram are ( -8 , -4 ) , ( 0 , -2 ) , ( 8 , 0 ) , ( 16 , 2 ) .
Each Input must have only one output in order to be the relation is a function ,
If ,The mapping diagram has the right to left representation or each input has more than one output , then the relation is not a function .
So , To have the relation as a function we must have only one output for one input.

Self-Assessment for Problem Solving
Solve each exercise. Then rate your understanding of the success criteria in your journal.

Question 9.
The mapping diagram represents the costs of reserving a hotel room for different numbers of nights.

a. Is the cost a function of the number of nights reserved?
b. Describe the relationship between the cost and the number of nights reserved.
Answer: a. Yes , The cost is a function of the number of nights reserved .
b. The relationship between the cost and the number of nights reserved is , For every night reservation of the room is increasing by $85 with increase in the next reservation ,

Explanation:
a. From the diagram we have ,
Ordered pairs are ( 1 , -$85 ) , ( 2 , $170 ) , ( 3 , $255 ) , ( 4 , $340 ) . each input has exactly one output ,
So , the relation is a function and ,
Yes , The cost is a function of the number of nights reserved .

b. The relationship between the cost and the number of nights reserved is ,
For every night reservation of the room is increasing by $85 with increase in the next reservation,
that is , input 1 = $85 as output
Input 2 = $85 + $85 = $170 as output
Input 3 = $170 + $85 = $255 as output
Input 2 = $255 + $85 = $340 as output

So, The relationship between the cost and the number of nights reserved is ,
For every night reservation of the room is increasing by $85 with increase in the next reservation.

Question 10.
DIG DEEPER!
The graph represents the number of contestants in each round of a talent competition.
a. Is the number of contestants a function of the round number?
b. Predict the number of contestants in the talent competition during Round 7. Explain your reasoning.

Answer: a. The number of contestants is a function of the round number.
b. The number of contestants in the talent competition during Round 7  are 2.

Explanation:
a. From the given graph , The ordered pairs are ( 1 , 128 ) , ( 2 , 64 ) , ( 3 , 32 ) , ( 4 , 16 ) .
Each input has only one output , The relation is a function .
So , the number of contestants is a function of the round number.

b. Firstly , The relation between the input and output is,
With every increase in the round number the number of contestants are decreasing by half the number of the previous round , That is, for  input 1 = 128 as output
For input 2 = 128 – 64 = 64  as output
For input 3 = 64 – 32 = 32  as output
For input 4 = 32 – 16 = 16  as output
For input 5 = 16 – 8 = 8  as output
For input 6 = 8 – 4 = 4  as output
For input 7 = 4 – 2 = 2  as output,
So, The number of contestants in the talent competition during Round 7  are 2 .

Relations and Functions Homework & Practice 7.1

Review & Refresh

Choose an appropriate data display for the situation. Explain your reasoning.
Question 1.
the number of runners in each
age group at a marathon
Answer: In a marathon ,the people of all age group are participating for a promotion on healthy lifestyle, The number of runners in each  group has kids, adults and old people to spread the awareness of leading a healthy life by running daily in the morning . Running or jogging in the morning can help us to maintain our body mass index at an optimal level which is good for heart. The Marathon is conducted by the government of health ministry to be example for the future generations.

Question 2.
the high temperature and the
attendance at a water park each day
Answer:  Generally, The water park is normally crowded depending on the season and the temperature, In summer the attendance in the waterpark is at the utmost point because of the high temperature and the seasonal vacation. Going to the water park in summer is super fun due to the number of  water slides , water rides will be a nice place to the whole family trip and as well as friends . In order to be there at a less crowded time spring is also a nice time to visit the water park .

Graph the linear equation.
Question 3.
y = 2x – 3
Answer:
Explanation:
Given , y = 2x – 3 , we know y = mx + c , where m = slope , c = constant
To obtain the graph , we should have ordered pairs ,
So , if x = 1 , then y = 2(1) – 3 = 2 – 3 = -1 . co-ordinates are (1 , -1)
if x = 2 , then y = 2(2) – 3 = 4 – 3 = 1 , co-ordinates are (2 , 1)
The co-ordinates (1 , -1) , (2 , 1) form a straight line .
So, y = 2x – 3 is a linear equation.

Question 4.
y = – 0.5x
Answer:
Explanation:
Given , y = -0.5x , we know y = mx + c , where m = slope , c = constant
To obtain the graph , we should have ordered pairs ,
So , if x = 0 , then y = -0.5(0) = 0 . co-ordinates are (0 , 0)
if x = 2 , then y = -0.5(2) = -1 , co-ordinates are (2 , -1)
The co-ordinates (0 , 0) , (2 , -1) form a straight line .
So, y = -0.5x is a linear equation.

Question 5.
y = – 3x + 4
Answer:
Explanation:
Given , y = – 3x + 4 , we know y = mx + c , where m = slope , c = constant
To obtain the graph , we should have ordered pairs ,
So , if x = 0 , then y = – 3(0) + 4 = 4 . co-ordinates are (0 , 4)
if x = 1 , then y = – 3(1) + 4 = -3 + 4 = 1 , co-ordinates are (1 , 1)
if x = 2 , then y = – 3(2) + 4 = -6 + 4 = -2 , co-ordinates are (2 , -2)
The co-ordinates (0 , 4) , (1 , 1) , (2 , -2) form a straight line .
So, y = – 3x + 4 is a linear equation.

Question 6.
Which word best describes two figures that have the same size and the same shape?
A. congruent
B. adjacent
C. parallel
D. similar
Answer:  A. congruent

Explanation:
Two figures which have the same size and shape are congruent.

Concepts, Skills, &Problem Solving

INTERPRETING DIAGRAMS Describe the relationship between the inputs and outputs in the diagram. Then complete the diagram. Is there more than one possible answer? Explain your reasoning. (See Exploration 1, p. 275.)
Question 7.

Answer: The relationship between the inputs and outputs in the diagram is ,
For every increase in number of input is having the output of adding -4 to the previous output.

Explanation:
The relationship between the inputs and outputs in the diagram is ,
For every increase in number of input is having the output of adding -4 to the previous output ,
for input 1 = -1 as output
for input 2 = -1 + (-4) = -5 as output
for input 3 = -5 + (-4) = -9 as output
for input 4 = -9 + (-4) = -13 as output
for input 5 = -13 + (-4) = -17 as output
for input 6 = -17 + (-4) = -21 as output.
So, The relationship between the inputs and outputs in the diagram is ,
For every increase in number of input is having the output of adding -4 to the previous output .

In this case , we are witnessing only one output for one input.

Question 8.

Answer: The relationship between the inputs and outputs in the diagram is,
Each input has the sports name and the output has the Starting letter of sports name.

Explanation:
The relationship between the inputs and outputs in the diagram is,
Each input has the sports name and the output has the Starting letter of sports name.
For input basketball = b as output
For input baseball = b as output
For input football = f as output
For input soccer = s as output
For input swimming = s as output,
So, The relationship between the inputs and outputs in the diagram is,
Each input has the sports name and the output has the Starting letter of sports name.

In this case we have more than one output for input.

LISTING ORDERED PAIRS List the ordered pairs shown in the mapping diagram.
Question 9.

Answer: Ordered pairs are ( 0 , 4 ) , ( 3 , 5 ) , ( 6 , 6 ) , ( 9 , 7 ) .

Explanation:
As shown , Ordered pairs are the combinations of input and output
So , Ordered pairs are ( 0 , 4 ) , ( 3 , 5 ) , ( 6 , 6 ) , ( 9 , 7 ) .

Question 10.

Answer: Ordered pairs are ( 1 , 8 ) , ( 3 , 8 ) , ( 3 , 4 ) , ( 5 , 6 ) , ( 7 , 2 ).

Explanation:
As shown , Ordered pairs are the combinations of input and output
So , Ordered pairs are ( 1 , 8 ) , ( 3 , 8 ) , ( 3 , 4 ) , ( 5 , 6 ) , ( 7 , 2 ).

Question 11.

Answer: Ordered pairs are ( 6 , -5 ) , ( 7 , -5 ) , ( 8 , -10 ) , ( 9 , -10 ).

Explanation:
As shown , Ordered pairs are the combinations of input and output
So , Ordered pairs are ( 6 , -5 ) , ( 7 , -5 ) , ( 8 , -10 ) , ( 9 , -10 ).

IDENTIFYING FUNCTIONS Determine whether the relation is a function.
Question 12.

Answer: The relation is not a function .

Explanation:
The each input has  more than two outputs , That is one input has multiple number of outputs.
Here , input 0 has two outputs which are 10 and 20 .
So , The relation is not a function .

Question 13.

Answer: The relation is a function .

Explanation:
Each input has exactly one output ,
So , The relation is a function .

Question 14.

Answer: The relation is a function .

Explanation:
Each input has exactly one output ,
So , The relation is a function .

Question 15.
YOU BE THE TEACHER
Your friend determines whether the relation shown in the mapping diagram is a function. Is your friend correct? Explain your reasoning.

Answer: The relation is not a function .

Explanation:
The each input has  more than two outputs , That is one input has multiple number of outputs.
Here , input 4 has four outputs which are 5, 6 , 7 and 8.
So , The relation is not a function .

REASONING Draw a mapping diagram that represents the relation. Then determine whether the relation is a function. Explain.
Question 16.

Answer:  The mapping diagram representing the relation is

Explanation:
From the given graph , co-ordinates of the ordering pairs are( 1 , 1 ), ( 3 , 3 ), ( -1 , -1 ), ( -3 , -3 ).
Each input has exactly one output ,
So , The relation is a function .

Question 17.

Answer: The mapping diagram representing the relation is

Explanation:
From the given graph , co-ordinates of the ordering pairs are( 0 , 8 ),( 2 , 8 ),( 4 , 8 ),( 6 , 8 ),( 8 , 8 ),( -2 , 8 ),( -4 , 8 ). Each input has exactly one output ,
So , The relation is a function.

Question 18.

Answer: The mapping diagram representing the relation is

Explanation:
From the given graph , co-ordinates of the ordering pairs are( -2 , 1 ),( -2 , 2 ),( -2 , 3 ),( -2 , 4 ),( -2 , 5 ),( -2 , 6 ).
Each input has more than one output ,
So , The relation is not a function.

Question 19.
MODELING REAL LIFE
The normal pressure at sea level is 1 atmosphere of pressure(1 ATM). As you dive below sea level, the pressure changes. The mapping diagram represents the pressures at different depths.

a. Complete the mapping diagram.
b. Is pressure a function of depth?
c. Describe the relationship between pressure and depth.
d. List the ordered pairs. Then plot the ordered pairs in a coordinate plane. What do you notice about the points?
e. RESEARCH What are common depths for beginner scuba divers? What are common depths for experienced scuba divers?
Answer: The detailed explanation of  all the answers are given below .

Explanation:
a. The mapping diagram is
b. Yes , the pressure is a function of depth, Because depth is related to pressure in the given mapping diagram.

c. The relationship between pressure and depth is,
for every 10m increase in Depth of input there is an increase in 1 ATM pressure .

d. The ordered pairs are ( 0 , 1 ) , ( 10 , 2 ) , (20 , 3 ) , ( 30 , 4 ) , ( 40 , 5 ), ( 50 , 6 ).
The plot of the ordered pairs in a coordinate plane is

From the graph, we have seen that, if the depth of the diving of scuba drivers increases then the water pressure increases with increase in depth. So, the graph have straight line .

e. The common depths for beginner scuba divers is 30 feet to 60 feet or 9 to 18 meters ,
The common depths for experienced scuba divers is more than 60 feet or more than 18 meters .

Question 20.
DIG DEEPER!
The table shows the cost of purchasing 1, 2, 3, or 4 T-shirts from a souvenir shop.

a. Is the cost a function of the number of T-shirts purchased?
b. Describe the relationship between the cost and the number cost per T-shirt of T-shirts purchased. How does the change as you purchase more T-shirts?
Answer: The detailed explanation of  all the answers are given below .

Explanation:
a. Yes , The cost is a function of the number of T-shirts purchased, Because the cost of the purchased T-shirts is varying with the number of T-shirts purchased.

b. The relationship between the cost and the number cost per T-shirt of T-shirts purchased is,
Input is the cost of 1 T-shirt is $10 as output , Then for 2 T-shirts cost will be $20
If 2 T-shirts will be purchased at same time, cost will be decreased by $2 so it will be $10 + 8 = $18 for 2 T-shirts.
As per the single T-shirt cost , For 3 T-shirts will be $30,
So in the table given that 3 T-shirts will cost $24 , because it cost $18 + 6 = $24 for 3 T-shirts.
It goes same for 4 T-shirts , For 4 T-shirts will be $40, because it cost $24 + 4 = $28 for 4 T-shirts.

The change as you purchase more T-shirts is For every increase in purchase of the number of T-shirts is decrease in the cost of total T-shirts purchased.

Question 21.
REPEATED REASONING
The table shows the outputs for several inputs. Use two methods to predict the output for an input of 200.

Answer: The output for an input of 200 is 1025.

Explanation:
Method 1. The relation between inputs and outputs is as follows,
y = 25 + 5x
As input increases by 1 , output increases by 5 units,
To find output of 200 as input ,
put x = 200 in the equation,
y = 25 + 5(200)
= 25 + 1000
= 1025.
So , y = 1025.

Method 2.  As the table shown, for every increase in input there is an increase in 5 numbers in output,
So , For 1 input = 25 + 5 = 30 as output
For 2 input = 30 + 5 = 35 as out put
For 3 input = 35 + 5 = 40 as out put
For 4 input = 40 + 5 = 45 as out put
By doing this for number 200 as input we have , 1025 as output.

Lesson 7.2 Representations of Functions

EXPLORATION 1

Using a Table to Describe Relationships
Work with a partner. Make a table that shows the relationship  between the figure number x and the area A of each figure. Then use an equation to find which figure has an area of 81 square units when the pattern continues.

Answer: a. The equation is y = 2x – 1, For figure has an area of 81 square units is 41.
b. The equation is y = x²,  For figure has an area of 81 square units is 9.

Explanation:
a. figure shows the 1 square unit of each box  for and it has a pattern of  2x – 1
figure 1 = 1 square unit
figure 2 =3 square units
figure 3 = 5 square unit and so on
So, the equation is y = 2x – 1 , it is in the form of y = mx + c,
Given to which figure has an area of 81 square units
substitute y  = 81, we have
y = 2x – 1
81 = 2x – 1
2x = 82
x = 41
So, For figure has an area of 81 square units is 41.

b. As shown above , we know that ,
figure 1 = 1 square unit
figure 2 =4 square units
figure 3 = 9 square unit and so on
Here we have a pattern of power of its own number,
So, the Equation  will be y = x²
Given to which figure has an area of 81 square units
substitute y  = 81, we have
x  = 9
So, For figure has an area of 81 square units is 9.

EXPLORATION 2

Using a Graph
Work with a partner. Use a graph to test the truth of each statement. If the statement is true, write an equation that shows how to obtain one measurement from the other.

a. “You can find the horsepower of a race-car car engine if you know its volume in cubic inches”

b. “You can find the volume of a race-car engine in cubic centimeters if you know its volume in cubic inches.”

Answer: a. Given ordered pairs are (200 , 375) , (350 , 650) , (350 , 250) , (500 , 600)
We can not find the horsepower of a race-car car engine if you know its volume in cubic inches
b. Given ordered pairs are (100 , 1640) , (200 , 3280) , (300 , 4920) ,
Yes, You can find the volume of a race-car engine in cubic centimeters if you know its volume in cubic inches

Explanation:
a. Given ordered pairs are (200 , 375) , (350 , 650) , (350 , 250) , (500 , 600)
We can not find the horsepower of a race-car car engine if you know its volume in cubic inches

b. Given ordered pairs are (100 , 1640) , (200 , 3280) , (300 , 4920) ,
Yes, You can find the volume of a race-car engine in cubic centimeters if you know its volume in cubic inches

Try It

Question 1.
Write a function rule for “The output is one-fourth of the input.”
Answer:  y = \(\frac{x}{4}\)

Explanation:
Let us say x is input and y is output , then
The output is one-fourth of the input, will be ,
y = \(\frac{x}{4}\).

Find the value of y when x = 5.
Question 2.
y = 4x – 1
Answer: y = 19.

Explanation:
Given, y = 4x – 1
substitute x = 5 , we get
y = 4(5) – 1
y = 20 – 1 = 19
So, y = 19.

Question 3.
y = 10x
Answer: y = 50

Explanation:
Given, y =10x
substitute x = 5 , we get
y = 10(5)
y = 50
So, y = 50.

Question 4.
y = 7 – 3x
Answer: y = -8.

Explanation:
Given, y = 7 – 3x
substitute x = 5 , we get
y = 7 – 3(5)
y = 7 – 15 = -8
So, y = -8.

Graph the function.
Question 5.
y = x + 1
Answer:

Explanation:
Given , y = x + 1  , we know y = mx + c , where m = slope , c = constant
To obtain the graph , we should have ordered pairs ,
So , if x = 0 , then y = 0 + 1 = 1 . co-ordinates are (0 , 1)
if x = 1 , then y = 1 + 1 = 2 . co-ordinates are (1 , 2)
if x = 2 , then y = 2 + 1 = 3 , co-ordinates are (2 , 3)
The co-ordinates (0 , 1) , (1 , 2) , (2 , 3) form a straight line .

Question 6.
y = – 3x
Answer:

Explanation:
Given , y = – 3x  , we know y = mx + c , where m = slope , c = constant
To obtain the graph , we should have ordered pairs ,
So , if x = 0 , then y = -3(0) = 0 . co-ordinates are (0 , 0)
if x = 1 , then y = -3(1) = -3 . co-ordinates are (1 , -3)
if x = 2 , then y = -3(2) = -6 , co-ordinates are (2 , -6)
if x = 3 , then y = -3(3) = -9 , co-ordinates are (3 , -9)
The co-ordinates (0 , 0) , (1 , -3) , (2 , -6) ,(3 , -9) form a straight line .

Question 7.
y = 3x + 2
Answer:

Explanation:
Given , y = 3x + 2  , we know y = mx + c , where m = slope , c = constant
To obtain the graph , we should have ordered pairs ,
So , if x = 0 , then y =3(0) + 2 = 2 . co-ordinates are (0 , 2)
if x = 1 , then y = 3(1) + 2= 5 . co-ordinates are (1 , 5)
if x = 2 , then y =3(2) + 2 = 7 , co-ordinates are (2 , 7)
The co-ordinates (0 , 2) , (1 , 5) , (2 , 7) form a straight line .

Self-Assessment for Concepts & Skills
Solve each exercise. Then rate your understanding of the success criteria in your journal.

WRITING FUNCTION RULES Write a function rule for the statement.
Question 8.
The output is three times the input.
Answer: y = 3x

Explanation:
Let us say x is input and y is output , then
The output is three times the input. will be ,
So , y = 3x .

Question 9.
The output is eight more than one-seventh of the input.
Answer: y = 8 + \(\frac{x}{7}\) .

Explanation:
Let us say x is input and y is output , then
The output is eight more than one-seventh of the input., will be ,
So, y = 8 + \(\frac{x}{7}\) .

EVALUATING A FUNCTION Find the value of y when x = 5.
Question 10.
y = 6x
Answer: y = 30

Explanation:
Given, y = 6x
substitute x = 5 , we get
y = 6(5) =30
So, y = 30

Question 11.
y = 11 – x
Answer: y = 6

Explanation:
Given, y = 11 – x
substitute x = 5 , we get
y = 11 – 5 = 6
So, y = 6.

Question 12.
y = \(\frac{1}{5}\)x + 1
Answer:  y = 2.

Explanation:
Given, y = \(\frac{1}{5}\)x + 1
substitute x = 5 , we get
y = \(\frac{x}{5}\) + 1
y= \(\frac{5}{5}\) + 1
y = 1 + 1 = 2
So, y = 2 .

GRAPHING A FUNCTION Graph the function.
Question 13.
y = – 2x
Answer:

Explanation:
Given , y = – 2x  , we know y = mx + c , where m = slope , c = constant
To obtain the graph , we should have ordered pairs ,
So , if x = 0 , then y = – 2(0) = 0 . co-ordinates are (0 , 0)
if x = 1 , then y = – 2(1)= -2 . co-ordinates are (1 , -2)
if x = 2 , then y =- 2(2) = -4 , co-ordinates are (2 , -4)
if x = 3 , then y =- 2(3) = -6 , co-ordinates are (3 , -6)
The co-ordinates (0 , 0) , (1 , -2) , (2 , -4) , (3 , -6) form a straight line .

Question 14.
y = x – 3
Answer:

Explanation:
Given , y = x – 3 , we know y = mx + c , where m = slope , c = constant
To obtain the graph , we should have ordered pairs ,
So , if x = 0 , then y = 0 – 3 = -3 . co-ordinates are (0 , -3)
if x = 1 , then y = 1 – 3= -2 . co-ordinates are (1 , -2)
if x = 2 , then y = 2 – 3 = -1 , co-ordinates are (2 , -1)
if x = 3 , then y = 3 – 3 = 0 , co-ordinates are (3 , 0)
The co-ordinates (0 , -3) , (1 , -2) , (2 , -1) , (3 , 0) form a straight line .

Question 15.
y = 9 – 3x
Answer: 

Explanation:
Given , y = 9 – 3x , we know y = mx + c , where m = slope , c = constant
To obtain the graph , we should have ordered pairs ,
So , if x = 0 , then y = 9 – 3(0) = 9 . co-ordinates are (0 , 9)
if x = 1 , then y = 9 – 3(1) = 6 . co-ordinates are (1 , 6)
if x = 2 , then y = 9 – 3(2) = 3 , co-ordinates are (2 , 3)
if x = 3 , then y = 9 – 3(3) = 0 , co-ordinates are (3 , 0)
The co-ordinates (0 , 9) , (1 , 6) , (2 , 3) , (3 , 0) form a straight line .

Question 16.
DIFFERENT WORDS, SAME QUESTION
Which is different? Find “both” answers.

Answer: As mentioned in the explanation below  a & d , b & c are different .

Explanation:
Given ,
a. what output is 4 more than twice the input 3?
Let us say that , y is output and x is input and given as 3 ,
then, we have y = 4 + 2(3) = 10.
b. What output is twice the sum of the input 3 and 4?
Let us say that , y is output and x is input and given as 3 ,
then, we have y = 2( 3 + 4 ) = 14.
c. what output is the sum of 2 times the input 3 and 4?
Let us say that , y is output and x is input and given as 3 ,
then, we have y = 2( 3 + 4 ) = 14.
d. what output is 4 increased by twice the input 3?
Let us say that , y is output and x is input and given as 3 ,
then, we have y = 4 + 2(3) = 10.

So, a & d , b & c are different .

Self-Assessment for Problem Solving
Solve each exercise. Then rate your understanding of the success criteria in your journal.

Question 17.
The World Health Organization(WHO) suggests having 23 health-care workers for every 10,000 people. How many health-care workers are needed to meet the WHO suggestion for a population of 250,000 people? Justify your answer using a graph.

Answer: So, 575 health-care workers are needed to meet the WHO suggestion for a population of 250,000 people

Explanation:
Given, The World Health Organization(WHO) suggests having 23 health-care workers for every 10,000 people.
we need to find how many health-care workers are needed to meet the WHO suggestion for a population of 250,000 people,
For every 10,000 people we have 23 care takers
Then for 250,000 people we have
\(\frac{23 × 250,000}{10,000}\)
= 23 × 25
= 575
So, 575 health-care workers are needed to meet the WHO suggestion for a population of 250,000 people

Question 18.
DIG DEEPER!
A truck produces 22 pounds of carbon dioxide for every gallon of diesel fuel burned. The fuel economy of the truck is 18 miles per gallon. Write and graph a function that describes the relationship between carbon dioxide produced and distance traveled.
Answer: y = 22x + 18  is the linear equation

Explanation:
Given, A truck produces 22 pounds of carbon dioxide for every gallon of diesel fuel burned.
The fuel economy of the truck is 18 miles per gallon.
So, we have y = 22x + 18 is in the form of y = mx +c
To obtain the graph , we should have ordered pairs ,
So , if x = 0 , then y = 22(0) + 18 = 18 . co-ordinates are (0 , 18)
if x = 1 , then y =22(1) + 18= 40  . co-ordinates are (1 , 40)
if x = 2 , then y =22(2) + 18 = 62 , co-ordinates are (2 , 62)
if x = 3 , then y =22(3) + 18 = 84  , co-ordinates are (3 , 84)
The co-ordinates (0 , 18) , (1 , 40) , (2 , 62) , (3 , 84) form a straight line .
The graph is

Representations of Functions Homework & Practice 7.2

Review & Refresh

Determine whether the relation is a function. 
Question 1.

Answer:  The relation is a function .

Explanation:
Each input has exactly one output ,
So , The relation is a function .

Question 2.

Answer:  The relation is a function .

Explanation:
Each input has exactly one output ,
So , The relation is a function .

Question 3.

Answer: The relation is not a function .

Explanation:
The each input has  more than two outputs , That is one input has multiple number of outputs.
Here , input 2 has two outputs which are 0 and -4 .
So , The relation is not a function .

Find the slope of the line.
Question 4.

Answer: slope = 1.

Explanation:
By using the slope equation , we know that
Slope = \(\frac{change in y}{change in x}\) or
slope = \(\frac{▲y}{▲x}\)
From the graph we know that change in y or ▲y is change from -2 to -4 =2
change in x or ▲x is change from 1 to 3 = 2 ,
So, slope = \(\frac{▲y}{▲x}\)
slope = \(\frac{2}{2}\)
slope = 1.

Question 5.

Answer: slope = \(\frac{5}{2}\) .

Explanation:
By using the slope equation , we know that
Slope = \(\frac{change in y}{change in x}\) or
slope = \(\frac{▲y}{▲x}\)
From the graph we know that change in y or ▲y is change from -4 to 1 = 5
change in x or ▲x is change from -1 to -3 = 2 ,
So, slope = \(\frac{▲y}{▲x}\)
slope = \(\frac{5}{2}\) .

Question 6.

Answer:  slope = \(\frac{1}{3}\) .

Explanation:
By using the slope equation , we know that
Slope = \(\frac{change in y}{change in x}\) or
slope = \(\frac{▲y}{▲x}\)
From the graph we know that change in y or ▲y is change from -4 to -3 = 1
change in x or ▲x is change from 1 to 4 = 3 ,
So, slope = \(\frac{▲y}{▲x}\)
slope = \(\frac{1}{3}\) .

Concepts, Skills, & Problem Solving

USING A GRAPH Use a graph to test the truth of the statement. If the statement is true, write an equation that shows how to obtain one measurement from the other measurement. (See Exploration 2, p. 281.)

Question 7.
“You can find the weight of a cell phone in ounces if you know its screen size in inches.”

Answer: we can does not find the weight of a cell phone in ounces if you know its screen size in inches.

From the given table , Ordered pairs are (4 , 4) , (4.7 , 4.8) , (5 , 4.8) , (5.5 , 6.4)
First find the slope m of the line containing the two given points (4, 4) and (4.7, 4.8)
m = (y₂-y₁) / (x₂-x₁)
m= (4.8 – 4) / (4.7 – 4)
m = 0.8/0.7 .
So, we can does not find the weight of a cell phone in ounces if you know its screen size in inches.

Question 8.
“You can find the age of a child in years if you know the age of the child in months.”

Answer: YES, y = 0.08x + 0.04 is a linear equations

Explanation:
From the given table , Ordered pairs are (9 , 0.75) , (12 , 1) , (15 , 1.25) , (24 , 2)
First find the slope m of the line containing the two given points (12 ,1) and (24, 2)
m = (y₂-y₁) / (x₂-x₁)
m= (2 – 1) / (24 – 12)
m = 1/12
m = 0.08.
substitute the slope in the (12 ,1) to get point slope to form a line.
y-y₁ = m (x-x₁)
y – 1 = 0.08(x – 12)
y –1 = 0.08x – 0.96
y = 0.08x –0.96 + 1
y =0.08 x + 0.04
So, y = 0.08x + 0.04 is a linear equation

WRITING FUNCTION RULES Write a function rule for the statement.
Question 9.
The output is half of the input.
Answer: y = \(\frac{x}{2}\).

Explanation:
Let us say x is input and y is output , then
The output is half of the input, will be ,
y = \(\frac{x}{2}\).

Question 10.
The output is eleven more than the input.
Answer: y = x + 11

Explanation:
Let us say x is input and y is output , then
The output is eleven more than the input, will be ,
y = x + 11

Question 11.
The output is three less than the input.
Answer: y = x – 3

Explanation:
Let us say x is input and y is output , then
The output is three less than the input, will be ,
y = x – 3

Question 12.
The output is the cube of the input.
Answer: y = x³

Explanation:
Let us say x is input and y is output , then
The output is the cube of the input, will be ,
y = x³

Question 13.
The output is six times the input.
Answer: y = 6x

Explanation:
Let us say x is input and y is output , then
The output is six times the input, will be ,
y = 6x

Question 14.
The output is one more than twice the input.
Answer: y = 2x + 1

Explanation:
Let us say x is input and y is output , then
The output is one more than twice the input, will be ,
y = 2x + 1

EVALUATING A FUNCTION Find the value of y for the given value of x.
Question 15.
y = x + 5; x = 3
Answer: y = 8

Explanation:
Given, y = x + 5
substitute x = 3 , we get
y = 3 + 5
So, y = 8.

Question 16.
y = 7x; x = – 5
Answer:  y = -35.

Explanation:
Given, y = 7x
substitute x = -5 , we get
y = 7(-5)
So, y = -35.

Question 17.
y = 1 – 2x; x = 9
Answer: y = -17

Explanation:
Given, y = 1 – 2x
substitute x = 9 , we get
y = 1 – 2(9)
y = 1 – 18
So, y = -17.

Question 18.
y = 3x + 2; x = 0.5
Answer: y = 5.5

Explanation:
Given, y = 3x + 2
substitute x = 0.5 , we get
y = 3(0.5) + 2
y = 3.5 + 2
So, y = 5.5 .

Question 19.
y = 2x³; x = 3
Answer: y = 54

Explanation:
Given, y = 2x³
substitute x = 3 , we get
y = 2(3)³
y = 2 × 27 = 54
So, y = 54.

Question 20.
y = \(\frac{x}{2}\) + 9; x = – 12
Answer: y = 3

Explanation:
Given, y = \(\frac{x}{2}\) + 9
substitute x = -12 , we get
y = \(\frac{-12}{2}\) + 9
y = -6 + 9
So, y = 3 .

GRAPHING A FUNCTION Graph the function.
Question 21.
y = x + 4
Answer: 

Explanation:
Given , y = x + 4  , we know y = mx + c , where m = slope , c = constant
To obtain the graph , we should have ordered pairs ,
So , if x = 0 , then y = 0 + 4 = 4 . co-ordinates are (0 , 4)
if x = 1 , then y = 1 + 4 = 5 . co-ordinates are (1 , 5)
if x = 2 , then y = 2 + 4 = 6 , co-ordinates are (2 , 6)
if x = 3 , then y = 3 + 4 = 7 , co-ordinates are (3 , 7)
The co-ordinates (0 , 4) , (1 , 5) , (2 , 6) , (3 , 7) form a straight line .

Question 22.
y = 2x
Answer:

Explanation:
Given , y = 2x  , we know y = mx + c , where m = slope , c = constant
To obtain the graph , we should have ordered pairs ,
So , if x = 0 , then y = 2(0) = 0 . co-ordinates are (0 , 0)
if x = 1 , then y = 2(1) = 2 . co-ordinates are (1 , 2)
if x = 2 , then y = 2(2) = 4 , co-ordinates are (2 , 4)
if x = 3 , then y = 2(3) = 6 , co-ordinates are (3 , 6)
The co-ordinates (0 , 0) , (1 , 2) , (2 , 4) , (3 , 6) form a straight line .

Question 23.
y = – 5x + 3
Answer:

Explanation:
Given , y = – 5x + 3  , we know y = mx + c , where m = slope , c = constant
To obtain the graph , we should have ordered pairs ,
So , if x = 0 , then y =- 5(0) + 3 = 3 . co-ordinates are (0 , 3)
if x = 1 , then y = – 5(1) + 3 = -2 . co-ordinates are (1 , -2)
if x = 2 , then y = – 5(2) + 3 = -7 , co-ordinates are (2 , -7)
if x = 3 , then y = – 5(3) + 3 = -12 , co-ordinates are (3 , -12)
The co-ordinates (0 , 3) , (1 , -2) , (2 , -7) , (3 , -12) form a straight line .

Question 24.
y = \(\frac{x}{4}\)
Answer:

Explanation:
Given , y = \(\frac{x}{4}\) , we know y = mx + c , where m = slope , c = constant
To obtain the graph , we should have ordered pairs ,
So , if x = 0 , then y = \(\frac{0}{4}\) = 0 . co-ordinates are (0 , 0)
if x = 1 , then y = \(\frac{1}{4}\) = 0.25 . co-ordinates are (1 , 0.25)
if x = 2 , then y = \(\frac{2}{4}\) = 0.5 , co-ordinates are (2 , 0.5)
if x = 3 , then y = \(\frac{3}{4}\) = 0.75 , co-ordinates are (3 , 0.75)
The co-ordinates (0 , 0) , (1 , 0.25) , (2 , 0.5) , (3 , 0.75) form a straight line .

Question 25.
y = \(\frac{3}{2}\)x + 1
Answer:

Explanation:
Given , y = \(\frac{3}{2}\)x + 1  , we know y = mx + c , where m = slope , c = constant
To obtain the graph , we should have ordered pairs ,
So , if x = 0 , then y =\(\frac{3}{2}\)(0) + 1 = 1 . co-ordinates are (0 , 1)
if x = 1 , then y = \(\frac{3}{2}\)(1) + 1= 2.5 . co-ordinates are (1 , 2.5)
if x = 2 , then y = \(\frac{3}{2}\)(2) + 1 = 4 , co-ordinates are (2 , 4)
if x = 3 , then y = \(\frac{3}{2}\)(3) + 1 = 5.5 , co-ordinates are (3 , 5.5)
The co-ordinates (0 , 1) , (1 , 2.5) , (2 , 4) , (3 , 5.5) form a straight line .

Question 26.
y = 1 + 0.5x
Answer:

Explanation:
Given , y = 1 + 0.5x  , we know y = mx + c , where m = slope , c = constant
To obtain the graph , we should have ordered pairs ,
So , if x = 0 , then y =1 + 0.5(0) = 1 . co-ordinates are (0 , 1)
if x = 1 , then y = 1 + 0.5(1) = 1.5 . co-ordinates are (1 , 1.5)
if x = 2 , then y = 1 + 0.5(2) = 2 , co-ordinates are (2 , 2)
if x = 3 , then y = 1 + 0.5(3) = 2.5 , co-ordinates are (3 , 2.5)
The co-ordinates (0 , 1) , (1 , 1.5) , (2 , 2) , (3 , 2.5)  form a straight line .

MATCHING Match the graph with the function it represents.
A. y = \(\frac{x}{3}\)
B. y = x + 1
C. y = – 2x + 6
Question 27.

Answer:  B. y = x + 1.

Explanation:
Given , y = x + 1  , we know y = mx + c , where m = slope , c = constant
To obtain the graph , we should have ordered pairs ,
So , if x = 0 , then y = 0 + 1 = 1 . co-ordinates are (0 , 1)
if x = 1 , then y = 1 + 1 = 2 . co-ordinates are (1 , 2)
if x = 2 , then y = 2 + 1 = 3 , co-ordinates are (2 , 3)
The co-ordinates (0 , 1) , (1 , 2) , (2 , 3) form a straight line .

Question 28.

Answer: c. y = – 2x + 6

Explanation:
Given , y = – 2x + 6  , we know y = mx + c , where m = slope , c = constant
To obtain the graph , we should have ordered pairs ,
So , if x = 0 , then y = – 2(0) + 6 = 6 . co-ordinates are (0 , 6)
if x = 1 , then y = – 2(1) + 6 = 4 . co-ordinates are (1 , 4)
if x = 2 , then y = – 2(2) + 6 = 2 , co-ordinates are (2 , 2)
if x = 3 , then y = – 2(3) + 6 = 0 , co-ordinates are (3 , 0)
The co-ordinates (0 , 6) , (1 , 4) , (2 , 2) , (3 , 0) form a straight line .

Question 29.

Answer: A. y = \(\frac{x}{3}\)

Explanation:
Given , y =  \(\frac{x}{3}\) , we know y = mx + c , where m = slope , c = constant
To obtain the graph , we should have ordered pairs ,
So , if x = 0 , then y =\(\frac{0}{3}\) = 0 . co-ordinates are (0 , 0)
if x = 1 , then y = \(\frac{1}{3}\)= 0.3 . co-ordinates are (1 , 0.3)
if x = 2 , then y = \(\frac{2}{3}\)= 0.6 , co-ordinates are (2 , 0.6)
The co-ordinates (0 , 0) , (1 , 0.3) , (2 , 0.6) form a straight line .

Question 30.
YOU BE THE TEACHER
Your friend graphs the function represented by the input-output table. Is your friend correct? Explain your reasoning.

Answer: Yes , He is correct

Explanation:

Ordered pairs are (-1 , -4) , (1 , -2) , (3 ,0) , (5 , 2)
these points form a straight line when graphed.
Yes , He is correct

Question 31.
MODELING REAL LIFE
A dolphin eats 30 pounds of fish per day.

a. Write and graph a function that relates the number p of pounds of fish that a dolphin eats in d days.
b. How many total pounds of fish does a dolphin eat in 30 days?
Answer:

Explanation:
a. Given , A dolphin eats 30 pounds of fish per day.
by each passing day eating fish is increased by the day passes .
So, y = 30x is the function,
The graph represents the function as

b. Given , A dolphin eats 30 pounds of fish per day.
then for 30 days ,
30 × 30 = 900 pounds
So, A dolphin eats 900 pounds of fish in 30 days

Question 32.
MODELING REAL LIFE
You fill a fish tank with 55 gallons of water on Saturday. The water evaporates at a rate of 1.5 gallons per day. You plan to add water when the tank reaches 49 gallons. When will you add water? Justify your answer.
Answer: As the action starts on Saturday , the tank will reach 49 gallons after 4 days , That is on Wednesday.

Explanation:
Given data ,, implies that slope of the function m = -1.5
The y intercept b= 55,
Then the equation  will be y = 55 – 1.5x
Given , You plan to add water when the tank reaches 49 gallons.
determine x for y = 49 ,
So, 49 = 55 – 1.5x ,
1.5x = 55 – 49
1.5x = 6
x = \(\frac{6}{1.5}\)
x = 4.

As the action starts on Saturday , the tank will reach 49 gallons after 4 days , That is on Wednesday.

USING AN EQUATION Find the value of x for the given value of y.
Question 33.
y = 5x – 7; y = – 22
Answer: x = -3

Explanation:
Given, y = 5x – 7
x = \(\frac{y + 7}{5}\)
substitute y = -22 , we get
x = \(\frac{-22 + 7}{5}\)
x = \(\frac{- 15}{5}\)
x = -3
So, x = -3 .

Question 34.
y = 9 – 7x; y = 37
Answer: x = -4

Explanation:
Given, y = 9 – 7x
x = \(\frac{9 – y}{7}\)
substitute y = 37 , we get
x = \(\frac{9 – 37}{7}\)
x = \(\frac{- 28}{7}\)
x = -4
So, x = -4 .

Question 35.
y = \(\frac{x}{4}\) – 7; y = 2
Answer: x = 36

Explanation:
Given, y = \(\frac{x}{4}\) – 7
x = 4( y + 7)
substitute y = 2 , we get
x = 4( 2 + 7)
x = 4(9)
x = 36
So, x = 36 .

Question 36.
PROBLEM SOLVING
You decide to make and sell bracelets. The cost of your materials is $84.00. You charge $3.50 for each bracelet.

a. P Write a function that represents the profit for selling b bracelets.
b. Which variable is independent? dependent? Explain.
c. You will break even when the cost of your materials equals your income. How many bracelets must you sell to break even?
Answer: a. A function that represents the profit for selling b bracelets is p = 3.5b – 84.
b. Here , the profit depends on the number of bracelets sold , b is the independent variable and p is the dependent variable.
c. To break even you must sell 24  bracelets.

Explanation:
a. Given , The cost of your materials is $84.00. You charge $3.50 for each bracelet,
Let p be the profit , b be the number of bracelets sold,
So, profit = income – cost .
p = 3.5b – 84.
Thus , A function that represents the profit for selling b bracelets is p = 3.5b – 84.

b. Here , the profit depends on the number of bracelets sold , b is the independent variable and p is the dependent variable.

c. set the income expression from part a equal to the cost of 84 and solve for b ,
So, income = cost .
3.5b = 84 ,
b = \(\frac{84}{3.5}\)
b = 24.

To break even you must sell 24  bracelets.

Question 37.
MODELING REAL LIFE
A furniture store is having a sale where everything is 40% off.
a. Write and graph a function that represents the amount of discount on an item at regular price.
b. You buy a bookshelf that has a regular price of $85. What is the sale price of the bookshelf?
Answer: a. The function is y = 0.4x and the graph is given below.
b. The sale price of the bookshelf s $51.

Explanation:
a. A function that represents the amount of discount on an item at regular price is ,
Given , 40% = 0.4 ,
To find the percent of the number , we should multiply the number by the percent in the decimal form ,
so, the equation is d = 0.4p ,
let us convert it in to a function form , y = 0.4x
we know y = mx + c , where m = slope , c = constant
To obtain the graph , we should have ordered pairs ,
So , if x = 0 , then y = 0.4(0) = 0 . co-ordinates are (0 , 0)
if x = 1 , then y = 0.4(1)= 0.4 . co-ordinates are (1 , 0.4)
if x = 2 , then y =0.4(2) = 0.8 , co-ordinates are (2 , 0.8)
if x = 3 , then y = 0.4(3) = 1.2 , co-ordinates are (3 , 1.2)
The co-ordinates (0 , 0) , (1 , 0.4) , (2 , 0.8) , (3 , 1.2) form a straight line .
The graph is
b. Given , You buy a bookshelf that has a regular price of $85.
The sale price of the bookshelf is ,
substituting the given price in p = 85 ,
it will be the discount d = 0.4 (85) = 34
Then the sale price is $85 – $34 = $51.

So, The sale price of the bookshelf s $51.

Question 38.
REASONING
You want to take a two-hour air boat tour. Which is a better deal, Snake Tours or Gator Tours? Use functions to justify your answer.

Answer: By using functions , $50 > $40 , So, Gator tours are cheaper than the snake tours .

Explanation:
Given , You want to take a two-hour air boat tour.
Let x be the hours of  air boat tour and y be the cost of air boat tour ,
Snake tours , y = 25x
putt x = 2 ,
So , y = 25 (2) = 50 .
y = 50.

Gator tour , y = 35 + \(\frac{5}{2}\)x
Put x = 2 ,
So, y = 35 + \(\frac{5}{2}\) x
y = 35 + 2.5x
y = 35 + 2.5 (2)
y = 35 + 5
y = 40 .

Finally $50 > $40 , So, Gator tours are cheaper than the snake tours

Question 39.
REASONING
The graph of a function is a line that passes through the points (3, 2), (5, 8), and (8, y). What is the value of y?
Answer: The value of y is 17 , so, The third given point is (8, 17)

Explanation:
First find the slope m of the line containing the two given points (3,2) and (5,8)
m = (y₂-y₁) / (x₂-x₁)
m= (8 – 2) / (5 – 3)
m = 6 / 2
m = 3
Then use the slope and one of the given points (3,2) to find the y-intercept
y = mx +
2 = 3(3) + b
2 = 9 + b
-7 = b
The equation is   y = 3x -7
Then find the third point (8, y) by replacing x by 8
y = 3x -7
y = 3(8) -7
y = 24 -7
y = 17

Question 40.
CRITICAL THINKING
Make a table where the independent variable is the side length of a square and the dependent variable is the perimeter. Make a second table where the independent variable is the side length of a square and the dependent variable is the area. Graph both functions in the same coordinate plane. Compare the functions.
Answer: The graph for the perimeter is linear , The graph for the Area is Quadratic .

Explanation:
Let us say , s be the side length of the square ,
Then the perimeter is P = 4s ,
The function will be y= 4x,
we know y = mx + c , where m = slope , c = constant
To obtain the graph , we should have ordered pairs ,
So , if x = 0 , then y = 4(0) = 0 . co-ordinates are (0 , 0)
if x = 1 , then y = 4(1) = 4 . co-ordinates are (1 , 4)
if x = 2 , then y = 4(2) =8 , co-ordinates are (2 , 8)
if x = 3 , then y = 4(3) = 0 , co-ordinates are (3 , 12)
The co-ordinates (0 , 0) , (1 , 4) , (2 ,8) , (3 , 12) form a straight line .

Table will be ,

Let us say , s be the side length of the square ,
Then the Area is A = s² ,
The function will be y=x²,
we know y = mx + c , where m = slope , c = constant
To obtain the graph , we should have ordered pairs ,
So , if x = 0 , then y = 0² = 0 . co-ordinates are (0 , 0)
if x = 1 , then y = 1² = 1 . co-ordinates are (1 , 1)
if x = 2 , then y = 2² =4 , co-ordinates are (2 , 4)
if x = 3 , then y = 3² = 9 , co-ordinates are (3 , 9)
The co-ordinates (0 , 0) , (1 , 1) , (2 ,4) , (3 , 9) form a straight line .

Second table is
Then the graph is 
The graph for the perimeter is linear , The graph for the Area is Quadratic .

Question 41.
PUZZLE
The blocks that form the diagonals of each square are shaded. Each block has an area of one square unit. Find the “green area” of Square 20. Find the “green area” of Square 21. Explain your reasoning.

Answer:  The green area of the Square 20 is 46 square units and The green area of the Square 21 is 48 square units.

Explanation:
Given , Each block has an area of one square unit,
Square 1 has  the diagonals of each square are shaded. the “green area” is 3 + 3 = 6 square units ,
Square 2 has  the diagonals of each square are shaded. the “green area” is 4 + 4 = 8 square units ,
Square 3 has  the diagonals of each square are shaded. the “green area” is 5 + 5 = 10 square units ,
Square 4 has  the diagonals of each square are shaded. the “green area” is 6 + 6 = 12 square units,
Square 5 has  the diagonals of each square are shaded. the “green area” is 7 + 7 = 14 square units ,
Here , The number of squares are increasing by one block with the square numbers.
So for the , Square 20 has  the diagonals of each square are shaded. the “green area” is 23 + 23 = 46 square units,
And Square 21 has  the diagonals of each square are shaded. the “green area” is 24 + 24 = 48 square units.

Lesson 7.3 Linear Functions

EXPLORATION 1

Writing and Graphing Functions
Work with a partner. Each table shows a familiar pattern from geometry.

• Determine what the variables x and y represent. Then write a function rule that relates y to x.
• Is the function a linear function? Explain your reasoning.

Answer: All of them are explained below

Explanation:
The variables x and y represents a rectangle
a. From the given table , Ordered pairs are (1 , 10) , (2 , 12) , (3 , 14) , (4 , 16)
First find the slope m of the line containing the two given points (1 ,-1) and (2, -2)
m = (y₂-y₁) / (x₂-x₁)
m= (-2 – (-1)) / (2 – 0)
m = -1 / 2 .
substitute the slope in the (1 , 10) to get point slope to form a line.
y-y₁ = m (x-x₁)
y – 10 = -1/2 ( x –1)
2(y – 10) = -x  + 1
2y – 20 = -x+ 1
2y = -x  + 21
y = \(\frac{-1}{2}\) (x – 21)
So ,  y = \(\frac{-1}{2}\) (x – 21) is linear function.

b. The variables x and y represent a circle
Ordered pairs are (1 , 3.14 ) , (2 , 6.28) , (3 , 9.42) , (4 , 12.5 )
Plot the points in the table , Draw a line through the points
First find the slope m of the line containing the two given points (1 , 3.14 ) , (2 , 6.28)
m = (y₂-y₁) / (x₂-x₁)
m= (6.28 –3.14) / (2– 1)
m = 3.14/1
m = 3.14
substitute the slope in the (8 , 4) to get point slope to form a line.
y-y₁ = m (x-x₁)
y – 3.14 =3.14 ( x –1)
y – 3.14 = 3.14x – 3.14
y = 3.14x – 3.14 + 3.14
y = 3.14x
So ,  y = 3.14x  is linear function.
Where x is the diameter of the circle.

c. The variables x and y represents a trapezoid
a. From the given table , Ordered pairs are (1 , 5) , (2 , 6) , (3 , 7) , (4 , 8)
First find the slope m of the line containing the two given points (1 ,5) and (2, 6)
m = (y₂-y₁) / (x₂-x₁)
m= (6 – 5) / (2 – 1)
m = 1 .
substitute the slope in the (1 ,5) to get point slope to form a line.
y-y₁ = m (x-x₁)
y – 5 = 1(x – 1)
y – 5 = x – 1
y = x – 1 + 5
y = x + 4
So, y = x + 4 is a linear equation

d. The variables x and y represents a cube
a. From the given table , Ordered pairs are (1 , 28) , (2 , 40) , (3 , 52) , (4 , 64)
First find the slope m of the line containing the two given points (1 ,28) and (2, 40)
m = (y₂-y₁) / (x₂-x₁)
m= (40 – 28) / (2 – 1)
m = 12 .
substitute the slope in the (1 ,28) to get point slope to form a line.
y-y₁ = m (x-x₁)
y – 28 = 12(x – 1)
y – 28 = 12x – 12
y = 12x – 12 + 28
y = 12x + 16
So, y = 12x + 16 is a linear equation.

Try It

Question 1.
Use the graph to write a linear function that relates y to x.

Answer: the linear function is y = \(\frac{-1}{2}\)x -1.

Explanation:
In order to write the function we have to write the ordered pairs of the graph ,
Ordered pairs are  (-4 , 1) , (-2 , 0 ) , (0 , -1) , ( 2, -2 )
First find the slope m of the line containing the two given points (0 ,-1) and (2, -2)
m = (y₂-y₁) / (x₂-x₁)
m= (-2 – (-1)) / (2 – 0)
m = -1 / 2 .
Because the line crosses the y axis at ( 0, -1 ) , The y intercept is -1.
So , the linear function is y = \(\frac{-1}{2}\)x -1.

Question 2.
Use the table to write a linear function that relates y to x.

Answer: the linear function is y = (0)x + 2.

Explanation:
Ordered pairs are (-2 , 2) , (-1 , 2) , (0 , 2) , (1 , 2)
Plot the points in the table , Draw a line through the points
First find the slope m of the line containing the two given points (0 ,2) and (1, 2)
m = (y₂-y₁) / (x₂-x₁)
m= (2 – 2) / (1 – 0)
m = 0
Because the line crosses the y axis at ( 0, 2 ) , The y intercept is 2.
So , the linear function is y = (0)x + 2.

Question 3.
WHAT IF?
The rate of descent doubles. Repeat parts (a) and (b).
Answer: a. the linear function is y = -1x + 65.
b. The slope indicates that the height decreases 1000 feet per minute.
The y intercept indicates that the descent begins at a cruising altitude of 65,000 feet.

Explanation:
a. From the Given table , The rate of descents is 5
If it doubles , then The rate of descents is 10.
The the ordered pairs will be (0 , 65) , (10 ,55) , (20 , 45) .
First find the slope m of the line containing the two given points (0 ,65) and (10, 55)
m = (y₂-y₁) / (x₂-x₁)
m= (55 – 65) / (10 – 0)
m = -10 / 10
m = -1
Because the line crosses the y axis at ( 0, 65 ) , The y intercept is 65.
So , the linear function is y = -1x + 65.

b. The slope indicates that the height decreases 1000 feet per minute.
The y intercept indicates that the descent begins at a cruising altitude of 65,000 feet.

Self-Assessment for Concepts & Skills
Solve each exercise. Then rate your understanding of the success criteria in your journal.

Question 4.
WRITING A LINEAR FUNCTION
Use the graph to write a linear function that relates y to x.

Answer:  The linear function is y = -4x -2 .

Explanation:
In order to write the function we have to write the ordered pairs of the graph ,
Ordered pairs are  (-2 , 6) , (-1 , 2 ) , (0 , -2) , ( 1, -6 )
First find the slope m of the line containing the two given points (0 ,-2) and (1, -6)
m = (y₂-y₁) / (x₂-x₁)
m= (-6 – (-2)) / (1 – 0)
m = -4 .
Because the line crosses the y axis at ( 0, -2) , The y intercept is -2.
So , the linear function is y = -4x -2 .

Question 5.
INTERPRETING A LINEAR FUNCTION
The table shows the revenue R (in millions of dollars) of a company when it spends A (in millions of dollars) on advertising.

a. Write and graph a linear function that relates R to A.
b. Interpret the slope and the y-intercept.
Answer:  a. The linear function is y = 2x + 2. and the graph is shown below
b. The slope indicates that the increasing in the amount of spending on advertising by 2 million dollars
The y intercept indicates that the Revenue begins to increasing from the 2 million dollars.

Explanation:
a. From the given table ,
The the ordered pairs will be (0 , 2) , (2 ,6) , (4 , 10) , (6 , 14) , (8 ,18) .
The graph is
First find the slope m of the line containing the two given points (0 ,2) and (2, 6)
m = (y₂-y₁) / (x₂-x₁)
m= (6 – 2) / (2 – 0)
m = 4 / 2
m = 2
Because the line crosses the y axis at ( 0, 2 ) , The y intercept is 2.
So , the linear function is y = 2x + 2.

b. The slope indicates that the increasing in the amount of spending on advertising by 2 million dollars
The y intercept indicates that the Revenue begins to increasing from the 2 million dollars.

Self-Assessment for Problem Solving
Solve each exercise. Then rate your understanding of the success criteria in your journal.

Question 6.
Manager A earns $15 per hour and receives a $50 bonus. The graph shows the earnings of Manager B. (a) Which manager has a greater hourly wage? (b) After how many hours does Manager B earn more money than Manager A?

Answer: a. Manager B has the greater hourly wage than Manager A .
b. As manager A receives a $50 bonus , Manager B has to work an hour extra to earn more money than Manager A .

Explanation:
a. Manager A earns $15 per hour and receives a $50 bonus.
The ordered pairs will be  (0 , 0) , (1 , 15) , (2 , 30) , (3 , 45)
The graph shows the earnings of Manager B.
Ordered pairs from the graph are  (0 , 0) , (1 , 25) , (2 , 50) , (3 , 75)
So, Manager B has the greater hourly wage than Manager A .

b. As manager A receives a $50 bonus , Manager B has to work an hour extra to earn more money than Manager A .

Question 7.
Each month, you start with 2 gigabytes of data and use 0.08 gigabyte per day. The table shows the amount (in gigabytes) of data that your friend has left days after the start of each month. Who runs out of data first? Justify your answer.

Answer:  you will be run out of data first

Explanation:
a. Given , Each month, you start with 2 gigabytes of data and use 0.08 gigabyte per day.
Let x be the number of days and y be the total data in gigabytes.
So, y = -0.08x + 2 ,
You will be out of data if , -0.08x + 2 = 0 ,
-0.08x + 2 = 0
2 = 0.08x
x = \(\frac{2}{0.08}\)
x = 25.
Hence ,you will be run out of data in 25 days.
b. Daily data usage for the friend will be given by the slope of the graph.
The the ordered pairs will be (0 , 3) , (7 ,2.3) , (14 , 1.6) .
First find the slope m of the line containing the two given points (7 ,2.3) and (14, 1.6)
m = (y₂-y₁) / (x₂-x₁)
m= (1.6 – 2.3) / (14 – 7)
m = -0.7 / 7
m = -0.1
Because the line crosses the y axis at ( 0, 3 ) , The y intercept is 3.
So , the linear function is y = -0.1x + 3.
Your friend will be out of data if ,
-0.1x + 3 = 0
3 = 0.1x
x = \(\frac{3}{0.1}\)
x = 30 .
Hence ,Friend will be run out of data in 30 days

So , you will be run out of data first

Linear Functions Homework & Practice 7.3

Review & Refresh

Write a function rule for the statement. Then graph the function.
Question 1.
The output is ten less than the input.
Answer: y = x – 10.

Explanation:
Let us say x is input and y is output , then
The output is ten less than the input, will be ,
y = x – 10.

Question 2.
The output is one-third of the input.
Answer: y = \(\frac{x}{3}\)

Explanation:
Let us say x is input and y is output , then
The output is one-third of the input, will be ,
y = \(\frac{x}{3}\) .

Solve the system.
Question 3.
y = x + 5
y = – 3x + 1
Answer: X = 0 , Y = 5

Explanation:
$Y=-3X+5$ ——————-(1)
$Y=X+5$ ——————(2)
Substitute $Y=X+5$ in equation (1)
X+5=−3X+5
Solve it for X
X+3X=5−5
4X=0
X=0/4=0
X = 0
Substitute $X=0$ in equation (1)
Y=0+5
Y=5

Question 4.
x + y = – 4
6x + 2y = 4
Answer:  X = 3 , $\mathrm{Y=\; -7\; .}$

Explanation:
$\mathrm{2Y}=-6X+4$ ——————-(1)
$Y=\; –\mathrm{X-4}$ ——————(2)
Substitute $Y=\; –\mathrm{X-4}$ in equation (1)
$\mathrm{2Y}=-6X+4$
2 ( $–\mathrm{X\; –\; 4\; )\; =\; -6X\; +\; 4\; -2X\; –\; 8\; =\; -6X\; +\; 4\; 6X\; -2X\; =\; 8\; +\; 4\; 4X\; =\; 12\; X\; =\; 3\; Substitute X=3 in\; equation\; (2)\; Y=–\; 3\; –\; 4\; Y=\; -7\; .}$

Question 5.
– 4x + 3y = 14
y = 2x + 8
Answer:  X = -5 , $Y\; =\; -2\; .$

Explanation:
$\mathrm{3Y}=\; 4X+14$ ——————-(1)
Y = 2X + 8 ——————(2)
Substitute $Y=\; 2X\; +\; 8$ in equation (1)
$\mathrm{3Y}=\; 4X+14$
3($2X\; +\; 8)\; =\; 4X+14\; 6X\; +\; 24\; =\; 4X\; +\; 14\; 6X\; –\; 4X\; =\; 14\; –\; 24\; 2X\; =\; -10\; X\; =\; -5\; Substitute X=\; -5 in\; equation\; (2)\; Y=\; 2(-5)\; +\; 8\; Y=\; -10\; +\; 8\; Y\; =\; -2.$

Concepts, Skills, &Problem Solving

WRITING AND GRAPHING FUNCTIONS The table shows a familiar pattern from geometry. (a) Determine what the variables x and y represent. Then write a function rule that relates y to x. (b) Is the function a linear function? Explain your reasoning. (See Exploration 1, p. 289.)
Question 6.

Answer: a. The variables x and y represent a right angle triangle
b. y = 2x  is linear function.

Explanation:
In order to write the function we have to write the ordered pairs
Ordered pairs are  (1 , 2) , (2 , 4) ,  (3 , 6 ) , (4 , 8), (5 , 10 ) .
a. the variables x and y represent a right angle triangle
Plot the points in the table , Draw a line through the points
First find the slope m of the line containing the two given points (1 , 2) , (2 , 4)
m = (y₂-y₁) / (x₂-x₁)
m= (4 – 2) / (2– 1)
m = 2/1
m = 2
b. substitute the slope in the (2 , 4) to get point slope to form a line.
y-y₁ = m (x-x₁)
y – 4 = 2 ( x – 2)
y – 4 = 2x – 4
y = 2x – 4 + 4
y = 2x
So ,  y = 2x  is linear function.

Given side of triangle is 4 then x= 4/2 = 2
x = 2 and y = 4.

Question 7.

Answer: y = 3.14x  is linear function. and The variables x and y represent a circle

Explanation:
Ordered pairs are (1 , 3.14 ) , (2 , 6.28) , (3 , 9.42) , (4 , 12.5 )
Plot the points in the table , Draw a line through the points
First find the slope m of the line containing the two given points (1 , 3.14 ) , (2 , 6.28)
m = (y₂-y₁) / (x₂-x₁)
m= (6.28 –3.14) / (2– 1)
m = 3.14/1
m = 3.14
substitute the slope in the (8 , 4) to get point slope to form a line.
y-y₁ = m (x-x₁)
y – 3.14 =3.14 ( x –1)
y – 3.14 = 3.14x – 3.14
y = 3.14x – 3.14 + 3.14
y = 3.14x
So ,  y = 3.14x  is linear function.
Where x is the diameter of the circle.

WRITING LINEAR FUNCTIONS Use the graph or table to write a linear function that relates y to x.
Question 8.

Answer: The linear function is y = \(\frac{4}{3}\)x +2

Explanation:
In order to write the function we have to write the ordered pairs of the graph ,
Ordered pairs are  (-3 , -2) , (0 , 2 ) , (3 , 6) , ( 6, 10 )
First find the slope m of the line containing the two given points (3 ,6) and (6, 10)
m = (y₂-y₁) / (x₂-x₁)
m= (10 – 6) / (6 – 3)
m = 4/3 .
Because the line crosses the y axis at ( 0, 2) , The y intercept is 2.
So , the linear function is y = \(\frac{4}{3}\)x +2 .

Question 9.

Answer: The linear function is y = (0)x +3 .

Explanation:
In order to write the function we have to write the ordered pairs of the graph ,
Ordered pairs are  (-2 , 3) , (-1 , 3 ) , (0 , 3) , ( 1, 3 ) , (2 , 3)
First find the slope m of the line containing the two given points (1 ,3) and (2, 3)
m = (y₂-y₁) / (x₂-x₁)
m= (3 – 3) / (2 – 1)
m = 0 .
Because the line crosses the y axis at ( 0, 3) , The y intercept is 3.
So , the linear function is y = (0)x +3 .

Question 10.

Answer: The linear function is y = \(\frac{-1}{4}\)x + 0.

Explanation:
Ordered pairs are (-8 , 2) , (-4 , 1) , (0 , 0) , (4 , -1)
Plot the points in the table , Draw a line through the points
First find the slope m of the line containing the two given points (-8 ,2) and (-4, 1)
m = (y₂-y₁) / (x₂-x₁)
m= (1 – 2) / (-4 – (-8))
m = -1/4
Because the line crosses the y axis at ( 0, 0 ) , The y intercept is 0.
So , the linear function is y = \(\frac{-1}{4}\)x + 0.

Question 11.

Answer: The linear function is y = \(\frac{2}{3}\)x + 5.

Explanation:
Ordered pairs are (-3 , 3) , (0 , 5) , (3 , 7) , (6 , 9)
Plot the points in the table , Draw a line through the points
First find the slope m of the line containing the two given points (3 ,7) and (6, 9)
m = (y₂-y₁) / (x₂-x₁)
m= (9 – 7) / (6 – 3)
m = 2/3
Because the line crosses the y axis at ( 0, 5 ) , The y intercept is 5.
So , the linear function is y = \(\frac{2}{3}\)x + 5.

Question 12.
INTERPRETING A LINEAR FUNCTION
The table shows the length y (in inches) of a person’s hair after x months.

a. Write and graph a linear function that relates y to x.
b. Interpret the slope and the y-intercept.
Answer: a. The linear function is y = 0.5x + 11.
b. The slope indicates that the increasing in the hair length
The y intercept indicates that the increasing in hair length by time.

Explanation:
a. Given ,
The ordered pairs will be (0 , 11) , (3 ,12.5) , (6 , 14) .
The graph is
First find the slope m of the line containing the two given points (3 ,12.5) and (6 , 14)
m = (y₂-y₁) / (x₂-x₁)
m= (14 – 12.5) / (6 – 3)
m = 1.5 / 3
m = 0.5
Because the line crosses the y axis at ( 0, 11 ) , The y intercept is 11.
So , the linear function is y = 0.5x + 11.

b. The slope indicates that the increasing in the hair length
The y intercept indicates that the increasing in hair length by time.

Question 13.
INTERPRETING A LINEAR FUNCTION
The table shows the percent (in decimal form) of battery power remaining x hours after you turn on a laptop computer.

a. Write and graph a linear function that relates y to x.
b. Interpret the slope, the x-intercept, and the y-intercept.
c. After how many hours is the battery power at75%?
Answer: a. The linear function is y = -0.2x + 1.
b. given below the explanation.
c. Battery will be 75% after 1.25 hours.

Explanation:
a. Given ,
The ordered pairs will be (0 , 1) , (2 ,0.6) , (4 , 0.2) .
The graph is
First find the slope m of the line containing the two given points (2 ,0.6) and (4 , 0.2)
m = (y₂-y₁) / (x₂-x₁)
m= (0.2 – 0.6) / (4 – 2)
m = -0.4 / 2
m = -0.2
Because the line crosses the y axis at ( 0, 1 ) , The y intercept is 1.
So , the linear function is y = -0.2x + 1.

b. Slope is -0.2 which means that as time increases by 1 hour, Battery power remaining decreases by 20% .
y intercept is 1, which means initially the battery power remaining before usage was 100%.
x intercept is 5 which means the battery remaining will be 0 after 5 hours.

c. battery percent will be 75% of 0.75 if ,
-0.2x + 1 = 0.75
0.2x = 1 – 0.75
x = 0.25/0.2
x = 1.25
Battery will be 75% after 1.25 hours.

Question 14.
MODELING REAL LIFE
The number y of calories burned after x minutes of kayaking is represented by the linear function y = 4.5x. The graph shows the number of calories burned by hiking.

a. Which activity burns more calories per minute?
b. You perform each activity for 45 minutes. How many total calories do you burn? Justify your answer.
Answer: a. hiking burns more calories than kayaking .
b. In kayaking, 202.5 calories are burnt per minute. and In hiking , 225 calories are burnt per minute.

Explanation:
a. The number y of calories burned after x minutes of kayaking is represented by the linear function y = 4.5x.
So, The ordered pairs of the graph are (0 , 0) , (1 , 4.5) , (2 , 9) , (3, 13.5)
Here , In kayaking burns 4.5 calories per minute .
For hiking ,
The ordered pairs of the graph are (0 , 0) , (1 , 5) , (2 , 10) , (3, 15)
Here , In hiking burns 5 calories per minute.
Thus , hiking burns more calories than kayaking .

b. Given , perform each activity for 45 minutes.
Liner function of the kayaking is y = 4.5x
substitute x = 45 in equation
y = 4.5 (45)
y = 202.5
In kayaking, 202.5 calories are burnt per minute.
Linear function of the hiking is y = 5x
substitute x = 45 in equation
y = 5 (45)
y = 225
In hiking , 225 calories are burnt per minute.

Question 15.
DIG DEEPER!
You and a friend race each other. You give your friend a 50-foot head start. The distance y (in feet) your friend runs after x seconds is represented by the linear function y = 14x + 50. The table shows your distance at various times throughout the race. For what distances will you win the race? Explain.

Answer: you will win the race for distances greater than 190 feet

Explanation:
The distance y (in feet) your friend runs after x seconds is represented by the linear function y = 14x + 50.
The slope of the line is 14 so , your friend runs at the rate of 14 ft per second
To find your rate , the ordered pairs are (2 , 38) , (4 , 76) , (6 , 114) , (8 , 152)
First find the slope m of the line containing the two given points (2 ,38) and (4 , 76)
m = (y₂-y₁) / (x₂-x₁)
m= (76 – 38) / (4 – 2)
m = 38 / 2
m = 19
You are running at the rate of 19 ft per second.
To get the linear equation , substitute the slope in the (2 , 38) to get point slope to form a line.
Then we have , y = 19x
Now if x = 10 , to run faster then ,
y = 19(10)
y = 190 .
Your friend linear equation is y = 14x + 50 .
if x = 10 ,then
y = 14(10) + 50
y = 140 + 50
y = 190.
So , for x > 10 , means you will run farther than your friend which means you would win the race .
Therefore, you will win the race for distances greater than 190 feet.

Question 16.
REASONING
You and your friend are saving money to buy bicycles that cost $175 each. You have $45 to start and save an additional $5 each week. The graph shows the amount y(in dollars) that your friend has after x weeks. Who can buy a bicycle first? Justify your answer.

Answer:  your friend will but the bicycle first.

Explanation:
Given , your friend savings are
the ordered pairs are (0,15) and (3,39)
First find the slope m of the line containing the two given points (0,15) and (3,39)
m = (y₂-y₁) / (x₂-x₁)
m= (39 – 15) / (3 – 0)
m = 24 / 3
m = 8
Because the line crosses the y axis at ( 0, 15 ) , The y intercept is 15.
So , the linear function is y = 8x + 15.
to buy bicycles that cost $175 each
if y = 175 , then
175 = 8x + 15
8x = 175 – 15
x = 160/8
x = 20
So, your friend need 20 weeks to buy the bicycle
Given, You have $45 to start and save an additional $5 each week
So , the linear function will be y = 5x + 45.
to buy bicycles that cost $175 each
if y = 175 , then
175 = 5x + 45
5x = 175 – 45
x = 130/5
x = 26
So, you need 26 weeks to buy the bicycle.
Hence, your friend will but the bicycle first.

Question 17.
CRITICAL THINKING
Is every linear equation a linear function? Explain your reasoning.
Answer: All linear equations produce straight lines when graphed, not all linear equations produce linear functions. In order to be a linear function, a graph must be both linear (a straight line) and a function (matching each x-value to only one y-value).

Question 18.
PROBLEM SOLVING
The heat index is calculated using the relative humidity and the temperature. For every 1 degree increase in the temperature from 94°F to 97°F at 75% relative humidity, the heat index rises 4°F. On a summer day, the relative humidity is 75%, the temperature is 94°F, and the heat index is 124°F. Estimate the heat index when the relative humidity is 75% and the temperature is 100°F. Use a function to justify your answer.

Answer:  Heat index is 148°F

Explanation:
The form of linear equation is y = mx + c
and the slope of the function is given by m = (y₂-y₁) / (x₂-x₁)
Let y be the heat index and x be the temperature
Given , (94, 124)
For every 1 degree increase in the temperature from 94°F to 97°F at 75% relative humidity, the heat index rises 4°F
that is m = 4
Since the line passes through (94, 124) means
124 = 4x + c
124 = 4(94) + c
124 = 376 + c
c = 124 – 376
c = -252
Linear function for the heat index is y = 4x – 252
put x = 100
So, y = 4(100) – 252
y = 400 – 252
y = 148.
Finally, Heat index is 148°F.

Lesson 7.4 Comparing Linear and Non Linear Functions

EXPLORATION 1

Comparing Functions
Work with a partner. Each equation represents the height h (in feet) of a falling object after t seconds.

• Graph each equation. Explain your method.
• Decide whether each graph represents a or function.
• Compare the falling objects.

Answer: Explained below

Explanation:
a. Given, h = 300 – 15t can be written as y = 300 – 15x
h = 300 – 15t  , we know y = mx + c , where m = slope , c = constant
To obtain the graph , we should have ordered pairs ,
So , if x = 0 , then y =300 – 15(0) = 300 . co-ordinates are (0 , 300)
if x = 1 , then y = 300 – 15(1) = 285 . co-ordinates are (1 , 285)
if x = 2 , then y = 300 – 15(2) = 270  , co-ordinates are (2 , 270)
if x = 3 , then y = 300 – 15(3) = 255  , co-ordinates are (3 , 255)
The co-ordinates (0 , 300) , (1 , 285) , (2 , 270) , (3 , 255) form a straight line .

The graph is

Given , h = 300- 16t²  , we know y = mx + c , where m = slope , c = constant
To obtain the graph , we should have ordered pairs ,
So , if x = 0 , then y =300- 16(0)²  = 300 . co-ordinates are (0 , 300)
if x = 1 , then y =300- 16(1)² = 284 . co-ordinates are (1 , 284)
if x = 2 , then y = 300- 16(2)² = 236 , co-ordinates are (2 , 236)
if x = 3 , then y = 300- 16(3)² = 252  , co-ordinates are (3 , 252)
The co-ordinates (0 , 300) , (1 , 284) , (2 , 236) , (3 , 252) does not form a straight line .

The graph is

b. For, h = 300 – 15t , The graph is linear so the so it is a function,
For h = 300- 16t² , The graph is linear so the so it is a function.

c. Sky diver has the slow fall while compared to the bowling ball , because parachute can be controlled with the wind and can be divert the destination point, and bowling ball cannot be controlled while falling.

Try It

Does the table represent a linear or nonlinear function? Explain.
Question 1.

Answer: y = 2x – 12 is linear function.

Explanation:
Ordered pairs are (2 , -8) , (4 , -4) , (6 , 0) , (8 , 4)
Plot the points in the table , Draw a line through the points
First find the slope m of the line containing the two given points (6 ,0) and (8, 4)
m = (y₂-y₁) / (x₂-x₁)
m= (4 – 0) / (8– 6)
m = 4/2
m = 2
substitute the slope in the (8 , 4) to get point slope to form a line.
y-y₁ = m (x-x₁)
y – 4 = 2 ( x – 8)
y – 4 = 2x – 16
y = 2x – 16 + 4
y = 2x – 12
So ,  y = 2x – 12 is linear function.

Question 2.

Answer: y = –\(\frac{5}{3}\)x + 25 is linear function.

Explanation:
Ordered pairs are (0 , 25) , (3 , 20) , (7 , 15) , (12 , 10)
Plot the points in the table , Draw a line through the points
First find the slope m of the line containing the two given points (0 ,25) and (3, 20)
m = (y₂-y₁) / (x₂-x₁)
m= (20 – 25) / (3– 0)
m = -5/3
Because the line crosses the y axis at ( 0, 25 ) , The y intercept is 25.
So , the linear function is y = –\(\frac{5}{3}\)x + 25.
So , y = –\(\frac{5}{3}\)x + 25 is linear function.

Does the equation represent a linear or nonlinear function? Explain.
Question 3.
y = x + 5
Answer: y = x + 5 is a linear function

Explanation:
Given , y = x + 5  , we know y = mx + c , where m = slope , c = constant
To obtain the graph , we should have ordered pairs ,
So , if x = 0 , then y = 0 + 5 = 5 . co-ordinates are (0 , 5)
if x = 1 , then y = 1 + 5 = 6 . co-ordinates are (1 , 6)
if x = 2 , then y = 2 + 5 = 7 , co-ordinates are (2 , 7)
The co-ordinates (0 , 5) , (1 , 6) , (2 , 7) form a straight line .
Each x input has only one y output so it is a function .
And it  forms a straight line when graphed .
So, y = x + 5 is a linear function.

Question 4.
y = \(\frac{4x}{3}\)
Answer: y = \(\frac{4x}{3}\) is a linear function.

Explanation:
Given , y = \(\frac{4x}{3}\) , we know y = mx + c , where m = slope , c = constant
To obtain the graph , we should have ordered pairs ,
So , if x = 0 , then y = \(\frac{4(0)}{3}\) = 0 . co-ordinates are (0 , 0)
if x = 1 , then y = \(\frac{4(1)}{3}\) = \(\frac{4}{3}\)  = 1.3. co-ordinates are (1 , 1.3)
if x = 2 , then y = \(\frac{4(2)}{3}\) = \(\frac{8}{3}\) = 2.6 , co-ordinates are (2 , 2.6)
The co-ordinates (0 , 0) , (1 ,1.3 ) , (2 , 2.6) form a straight line .
Each x input has only one y output so it is a function .
And it forms a straight line when graphed .
So, y = \(\frac{4x}{3}\) is a linear function.

Question 5.
y = 1 – x²
Answer: y = 1 – x² is not a linear function.

Explanation:
Given , y = 1 – x²  , we know y = mx + c , where m = slope , c = constant
To obtain the graph , we should have ordered pairs ,
So , if x = 0 , then y = 1 – 0² = 1 . co-ordinates are (0 , 1)
if x = 1 , then y = 1 – 1² = 0 . co-ordinates are (1 , 0)
if x = 2 , then y = 1 – 2² = -3 , co-ordinates are (2 , -3)
The co-ordinates (0 , 5) , (1 , 6) , (2 , 7) form a straight line .
Each x input has only one y output so it is a function .
And it does not forms a straight line when graphed .
So, y = 1 – x² is not a linear function.

Does the graph represent a linear or nonlinear function? Explain.
Question 6.

Answer: The graph represents a nonlinear function.

Explanation:
In order to write the function we have to write the ordered pairs of the graph ,
Ordered pairs are  (0 , 2) , (-1 , 0) ,  (-2 , -2 ) , (-3 , -4), (0 , 1 ) , (2 , -2) , ( 3, -4 )
The inputs have more than one output ,
And points form a straight line
So , the graph is non linear function

Question 7.

Answer: The graph is a linear function

Explanation:
In order to write the function we have to write the ordered pairs of the graph ,
Ordered pairs are  (0 , 0) , (-1 , -1) ,  (-2 , -2 ) , (-3 , -3), (1 , 1 ) , (2 , 2) , ( 3, 3 )
The inputs have exactly one output ,
And points form a straight line
So , the graph is a linear function.

Self-Assessment for Concepts & Skills
Solve each exercise. Then rate your understanding of the success criteria in your journal.

IDENTIFYING FUNCTIONS Does the table or graph represent a linear or nonlinear function? Explain.
Question 8.

Answer: It is not a linear function

Explanation:
Ordered pairs are (3 , 0) , (-1 , 2) , (-5 , 4) , (-9 , 6)
Each input has exactly one output
and it does not form a straight line when graphed
So, it is not a linear function .

Question 9.

Answer: The graph is non linear function

Explanation:
In order to write the function we have to write the ordered pairs of the graph ,
Ordered pairs are  (0 , -1) , (-1 , 0) ,  (-2 , 3 ) , (1 , 0 ) , (2 , 3) .
The inputs have exactly one output ,
And points does not form a straight line
So , the graph is non linear function

Question 10.
WHICH ONE DOESN’T BELONG?
Which equation does not belong with the other three? Explain your reasoning.

Answer: 5xy = -2 does not belong with the other three.

Explanation:
15y = 6x , y = \(\frac{2}{5}\)x , 10y = 4x .
These are evaluated as 5y = 2x
5xy = -2 , is different from 5y = 2x.

Self-Assessment for Problem Solving
Solve each exercise. Then rate your understanding of the success criteria in your journal.

Question 11.
The loudness of sound is measured in (dB). The graph shows the loudness y of a sound (in decibels) x meters from the source of the sound. Is the relationship between loudness and distance linear or nonlinear? Approximate the loudness of the sound 12 meters from the source.

Answer: The relationship between loudness and distance  is nonlinear Function. And the loudness of the sound 12 meters from the source is approximately 85dB as shown in the graph.

Explanation:
As shown in the graph , the plot of the points does not form a straight line ,
Its a parabolic decay , The amount of loudness decreases with the increase in distance,
So, The relationship between loudness and distance  is nonlinear Function.

And the loudness of the sound 12 meters from the source is approximately 85dB as shown in the graph.

Question 12.
A video blogger is someone who records a video diary. A new website currently hosts 90 video bloggers and projects a gain of 10 video bloggers per month. The table below shows the actual numbers of video bloggers. How does the projection differ from the actual change?

Answer: Projections are more than the actual values

Explanation:

So, Projections are more than the actual values

Comparing Linear and Non Linear Functions Homework & Practice 7.4

Review & Refresh

Write a linear function that relates y to x.
Question 1.

Answer: The linear function is y = x – 2

In order to write the function we have to write the ordered pairs of the graph ,
Ordered pairs are  (0 , -2) , (1 , -1 ) , (-1 , -3) , ( 2, 0), (3 , 1) , (4 , 2) , ( 5, 3)
First find the slope m of the line containing the two given points (2 ,0) and (3, 1)
m = (y₂-y₁) / (x₂-x₁)
m= (1 – 0) / (3 – 2)
m = 1 .
Because the line crosses the y axis at ( 0, -2 ) , The y intercept is -2.
So , the linear function is y = x – 2 .

Question 2.

Answer: The linear function is y =\(\frac{-1}{1.5}\)x + 5.

Explanation:
Ordered pairs are (0 , 5) , (1.5 , 4) , (3 , 3) , (4.5 , 2)
Plot the points in the table , Draw a line through the points
First find the slope m of the line containing the two given points (1.5 ,4) and (3, 3)
m = (y₂-y₁) / (x₂-x₁)
m= (3 – 4) / (3 – 1.5)
m = -1 /1.5
Because the line crosses the y axis at ( 0, 5 ) , The y intercept is 5.
So , the linear function is y =\(\frac{-1}{1.5}\)x + 5.

The vertices of a figure are given. Draw the figure and its image after a dilation with the given scale factor. Identify the type of dilation.
Question 3.
A (- 3, 1), B (- 1, 3), C (- 1, 1); k = 3
Answer: The New right angle triangle is larger than the original one So , its a increase .

Explanation:
Given , (- 3, 1),  (- 1, 3),  (- 1, 1) these pairs form a right angle triangle
K = 3 , For the dilation figure multiply the 3 with the given ordered pairs , then
(- 3, 1) × 3 = ( -9 , 3)
(- 1, 3) × 3 = ( -3 , 9)
(- 1, 1) × 3 = (-3 , 3)
From these new ordered pairs we form a new  right angle triangle
The figure is
The New right angle triangle is larger than the original one So , its a increase .

Question 4.
J (2, 4), K (6, 10), L (8, 10), M (8, 4); k = \(\frac{1}{4}\)
Answer: It is a reduction

Explanation:
Given , (2, 4),  (6, 10),  (8, 10) ,(8,4) these pairs forms a figure
K = 0.25 , For the dilation figure multiply the 3 with the given ordered pairs , then
(2, 4) × 0.25 = (0.5, 1)
(6, 10) × 0.25 = (1.5, 2.5)
(8, 10) × 0.25 = (2, 2.5)
(8, 4) × 0.25 = (2 , 1)
From these new ordered pairs we form a new figure
The figure is
The New figure is smaller than the original , So, It is a reduction .

Concepts, Skills, & Problem Solving

COMPARING FUNCTIONS Graph each equation. Decide whether each graph represents a linear or nonlinear function. (See Exploration 1, p. 295.)
Question 5.
h = 5 + 6t Equation 1
h = 5 + 6t² Equation 2
Answer: h = 5 + 6t Equation 1 is a linear function
h = 5 + 6t² Equation 2 is a non linear function .

Explanation:
Given , h = 5 + 6t  , we know y = mx + c , where m = slope , c = constant
To obtain the graph , we should have ordered pairs ,
So , if x = 0 , then y = 5 + 6(0) = 5 . co-ordinates are (0 , 5)
if x = 1 , then y = 5 + 6(1) = 11 . co-ordinates are (1 , 11)
if x = 2 , then y = 5 + 6(2) = 17 , co-ordinates are (2 , 17)
if x = 3 , then y = 5 + 6(3) = 23 , co-ordinates are (3 , 23)
The co-ordinates (0 , 5) , (1 , 11) , (2 , 17) , (3 , 23) form a straight line .

Given , h = 5 + 6t²  , we know y = mx + c , where m = slope , c = constant
To obtain the graph , we should have ordered pairs ,
So , if x = 0 , then y = 5 + 6(0)² = 5 . co-ordinates are (0 , 5)
if x = 1 , then y = 5 + 6(1)² = 11 . co-ordinates are (1 , 11)
if x = 2 , then y = 5 + 6(2)² = 26 , co-ordinates are (2 , 26)
if x = 3 , then y = 5 + 6(3)² = 59 , co-ordinates are (3 , 59)
The co-ordinates (0 , 5) , (1 , 11) , (2 , 26) , (3 , 59) does not form a straight line .

The graph of both equations is
So, h = 5 + 6t Equation 1 is a linear function
h = 5 + 6t² Equation 2 is a non linear function .

Question 6.
y = – \(\frac{x}{3}\) Equation 1
y = – \(\frac{3}{x}\) Equation 2
Answer:  y = – \(\frac{x}{3}\) Equation 1 is a linear function
y = – \(\frac{3}{x}\) Equation 2 is a non linear function.

Given , y =- \(\frac{x}{3}\) , we know y = mx + c , where m = slope , c = constant
To obtain the graph , we should have ordered pairs ,
So , if x = 0 , then y =- \(\frac{0}{3}\) = 0 . co-ordinates are (0 , 0)
if x = 1 , then y = – \(\frac{1}{3}\) = – 0.3 . co-ordinates are (1 , – 0.3 )
if x = 2 , then y = – \(\frac{2}{3}\) = – 0.6 , co-ordinates are (2 ,-0.6)
if x = 3 , then y = – \(\frac{3}{3}\) = -1 , co-ordinates are (3 , -1)
The co-ordinates (0 , 0) , (1 , -0.3) , (2 , -0.6) , (3 , -1) form a straight line .

Given , y =- \(\frac{3}{x}\) , we know y = mx + c , where m = slope , c = constant
To obtain the graph , we should have ordered pairs ,
So , if x = 0 , then y =- \(\frac{3}{0}\) = no number
if x = 1 , then y = – \(\frac{3}{1}\) = – 3 . co-ordinates are (1 , – 1 )
if x = 2 , then y = – \(\frac{3}{2}\) = – 1.5 , co-ordinates are (2 ,-1.5)
if x = 3 , then y = – \(\frac{3}{3}\) = -1 , co-ordinates are (3 , -1)
The co-ordinates (1 , -1) , (2 , -1.5) , (3 , -1) form a straight line .

The graph of both the equations is
So,  y = – \(\frac{x}{3}\) Equation 1 is a linear function
y = – \(\frac{3}{x}\) Equation 2 is a non linear function.

IDENTIFYING FUNCTIONS FROM TABLES Does the table represent a linear or nonlinear function? Explain.
Question 7.

Answer:  linear function is y = 4x + 4.

Explanation:
Ordered pairs are (0 , 4) , (1 , 8) , (2 , 12) , (3 , 16)
Plot the points in the table , Draw a line through the points
First find the slope m of the line containing the two given points (2 , 12) and (3 , 16)
m = (y₂-y₁) / (x₂-x₁)
m= (16 – 12) / (3– 2)
m = 4/1
m = 4
Because the line crosses the y axis at ( 0, 4 ) , The y intercept is 4.
So , the linear equation is y = 4x + 4.
And it is a linear function.

The graph is

Question 8.

Answer: y = 4x – 6 is linear function.

Explanation:
Ordered pairs are (6 , 21) , (5 , 15) , (4 , 10) , (3 , 6)
Plot the points in the table , Draw a line through the points
First find the slope m of the line containing the two given points (4 , 10) and (3 , 6)
m = (y₂-y₁) / (x₂-x₁)
m= (6 – 10) / (3– 4)
m = -4/-1
m = 4
substitute the slope in the(4 , 10) to get point slope to form a line.
y-y₁ = m (x-x₁)
y – 10 = 4 ( x – 4)
y – 10 = 4x – 16
y = 4x – 16 + 10
y = 4x – 6
So ,  y = 4x – 6 is linear function.

The graph is

IDENTIFYING FUNCTIONS FROM EQUATIONS Does the equation represent a linear or nonlinear function? Explain.
Question 9.
2x + 3y = 7
Answer: The function is linear when m = \(\frac{-2}{3}\) and c = \(\frac{7}{3}\)

Explanation:
Given ,2x + 3y = 7
3y = 7 – 2x
y = \(\frac{-2}{3}\)x+ \(\frac{7}{3}\)
So, The function is linear when m = \(\frac{-2}{3}\) and c = \(\frac{7}{3}\)

Question 10.
y + x = 4x + 5
Answer: The function is linear when m = 3 and c = 5 .

Explanation:
Given , y + x = 4x + 5
y = 4x – x + 5
y = 3x + 5
So, The function is linear when m = 3 and c = 5 .

Question 11.
y = \(\frac{8}{x^{2}}\)
Answer: The function is linear when m = 8 and c = 0 .

Explanation:
Given , y = \(\frac{8}{x^{2}}\)
slope m = 8
c = 0
So, The function is linear when m = 8 and c = 0 .

IDENTIFYING FUNCTIONS FROM GRAPHS Does the graph represent a linear or nonlinear function? Explain.
Question 12.

Answer: The graph is linear function

Explanation:
In order to write the function we have to write the ordered pairs of the graph ,
Ordered pairs are  (0 , 1) , (2 , 0) ,  (4 , -1 ) , (-2 , 2), ( -4, 3 )
The inputs have exactly one output ,
And points form a straight line
So , the graph is linear function

Question 13.

Answer: The graph is non linear function.

Explanation:
In order to write the function we have to write the ordered pairs of the graph ,
Ordered pairs are  (0 , 0) , (-1 , -1) ,  (-4 , -2 ) , (1 , 1), ( 4, 2 )
The inputs have exactly one output ,
And points does not form a straight line
So , the graph is non linear function

Question 14.
IDENTIFYING A FUNCTION
The graph shows the volume V (in cubic feet) of a cube with an edge length of x feet. Does linear nonlinear the graph represent a linear or nonlinear function? Explain.

Answer: The graph is non linear function

n order to write the function we have to write the ordered pairs of the graph ,
Ordered pairs are  (1 , 1) , (2 , 8) ,  (3 , 27 ) , (4 , 64)
The inputs have exactly one output ,
And points does not form a straight line
So , the graph is non linear function

Question 15.
MODELING REAL LIFE
The frequency y (in terahertz) of a light wave is a function of its wavelength x (in nanometers). Is the function relating the wavelength of light to its frequency linear or nonlinear?

Answer: The function is a non linear function

Explanation:
table is as follows
change in x is constant but change in y is not constant , it is increasing
So, the function is a non linear function .

Question 16.
DIG DEEPER!
The table shows the cost (in dollars) of pounds of sun flower seeds.

a. What is the missing -value that makes the table represent a linear function?
b. Write a linear function that represents the cost of x pounds of seeds. Interpret the slope.
c. Does the function have a maximum value? Explain your reasoning.
Answer:  a. 3 pounds = $4.2
b. y = 1.4x  is linear function.
c.  If y has maximum value then the x also has maximum value.

Explanation:
a. As per the table 1 pound = $1.4
2 pounds = $2.8
3pounds = $4.2
4 pounds = $5.6
So, the price is increasing with weight of the seeds.

b. Ordered pairs are (2 , 2.8) , (3 , 4.2) , (4 , 5.6)
Plot the points in the table , Draw a line through the points
First find the slope m of the line containing the two given points (3 , 4.2) and (4 , 5.6)
m = (y₂-y₁) / (x₂-x₁)
m= (5.6 – 4.2) / (4 – 3)
m = 1.4
substitute the slope in the (3 , 4.2) to get point slope to form a line.
y-y₁ = m (x-x₁)
y – 4.2 = 1.4 ( x – 3)
y – 4.2 = 1.4x – 4.2
y = 1.4x – 4.2 + 4.2
y = 1.4x
So ,  y = 1.4x  is linear function.

c. As shown in the table , and the function if y increases then x also increases with respect to the y
So, if y has maximum value then the x also has maximum value.

Question 17.
MODELING REAL LIFE
A birch tree is 9 feet tall and grows at a rate of 2 feet per year. The table shows the height h (in feet) of a willow tree after x years.

a. Does the table represent a linear or nonlinear function? Explain.
b. Which tree is taller after 10 years? Explain.
Answer: There is no linear relationship between x and y .

Explanation:
Table is as follows
Change in y is constant but change in x is increasing , not a constant
Hence, there is no linear relationship between x and y .

Question 18.
CRITICAL THINKING
In their first year, Show A has 7 million viewers and Show B has 5 million viewers. Each year, Show A has 90% of the viewers it had in the previous year. Show B loses 200,000 viewers each year.
a. Determine whether the function relating the year to the number of viewers is linear or nonlinear for each show.
b. Which show has more viewers in its sixth year?
Answer: a. The function relating the year to the number of viewers is linear
b. Both shows  has same number of viewers in the sixth year .

Explanation:
a. Given, In their first year, Show A has 7 million viewers and Show B has 5 million viewers. Each year, Show A has 90% of the viewers it had in the previous year. Show B loses 200,000 viewers each year.
For show A
So , In first year = 7
2 year = 90% of 7 = 6.3
3 year = 90% of 6.3 = 5.6
4 year = 90% of 5.6 = 5.04
5 year = 90% of 5 = 4.5
6 year = 90% of 4.5 = 4.05
So the ordered pairs are (1 , 7) , (2 , 6.3) , (3 , 5.6) , (4 , 5), (5 , 4.5) , (6 , 4)

For show B
In first year = 5 , As the viewers reduces by 2,00,000 in 5M
2 year = 5 – 0.2 = 4.8
3 year = 4.8 – 0.2 = 4.6
4 year = 4.6 – 0.2 = 4.4
5 year = 4.4 – 0.2 = 4.2
6 year = 54.2 – 0.2 = 4
So the ordered pairs are (1 , 5) , (2 , 4.8) , (3 , 4.6) , (4 , 4.4), (5 , 4.2) , (6 , 4)
As the year increases the viewers are also decreasing constantly as per the individual shows
So, The function relating the year to the number of viewers is linear .

b. As shown in part a , the ordered pairs having (6,4) represents the number of viewers to the year
So, Both shows  has same number of viewers in the sixth year .

Question 19.
NUMBER SENSE
The ordered pairs represent a function. (0,- 1), (1, 0), (2, 3), (3, 8), and (4, 15)
a. Graph the ordered pairs and describe the pattern. Is the function linear or nonlinear?
b. Write an equation that represents the function.
Answer: a. The graph is shown below and function is linear
b. The linear equation is y = 7x – 1.

Explanation:
Given, ordered pairs represent a function. (0,- 1), (1, 0), (2, 3), (3, 8), and (4, 15)
a. the graph is 
Each input has exactly one output and it forms a straight line So, the graph is linear
b. First find the slope m of the line containing the two given points (3 ,8) and (4, 15)
m = (y₂-y₁) / (x₂-x₁)
m= (15 – 8) / (4– 3)
m = 7/1
m = 7
Because the line crosses the y axis at ( 0, -1 ) , The y intercept is -1.
So , the linear equation is y = 7x – 1.

Lesson 7.5 Analyzing and Sketching Graphs

EXPLORATION 1

Matching Situations to Graphs
Work with a partner. Each graph shows your speed during a bike ride. Match each situation with its graph. Explain your reasoning.

a. You increase your speed, then ride at a constant speed along a bike path. You then slow down until you reach your friend’s house. Analyze Relationships
b. You increase your speed, then go down a hill. You then quickly come to a stop at an intersection.
c. You increase your speed, then stop at a store for a couple of minutes. You then continue to ride, increasing your speed.
d. You ride at a constant speed, then go up a hill. Once on top of the hill, you increase your speed.
Answer: a – C ,
b – A ,
c – D ,
d – B ,

Explanation:
a. You increase your speed, then ride at a constant speed along a bike path. You then slow down until you reach your friend’s house. The graph C has the perfect graph representing the situation of given question.
b. You increase your speed, then go down a hill. You then quickly come to a stop at an intersection.
Because The graph A has the bike speed representing the situation for the time .
c. You increase your speed, then stop at a store for a couple of minutes. You then continue to ride, increasing your speed. Thus, The graph D is the final answer for the question
d. You ride at a constant speed, then go up a hill. Once on top of the hill, you increase your speed.
Because of the speed with respect to time the graph B is the correct answer for the question.

EXPLORATION 2

Interpreting a Graph
Work with a partner. Write a short paragraph that describe show the height changes over time in the graph shown. What situation can this graph represent?

Answer: The Graph can be representing a situation for low and high tides of the Ocean

Explanation:
As shown in the figure, The graph is plotted between the height and time,
We can take an example of an Ocean for its waves , As the time passes at the morning of a normal day, The waves of the ocean start rising higher at a period of time, and for the time being maintaining a peak height then drops to a lower height at a particular intervals of time , this process takes place for a while and vise versa.
Thus, the Graph can be representing a situation for low and high tides of the Ocean

Try It

Question 1.
The graph shows the location of a pelican relative to your location.

a. Describe the path of the pelican.
b. Write an explanation for the decrease in the vertical distance of the pelican.
Answer: Both of them are explained below.

Explanation:
a. The path of the pelican is flying in the air , As they always fly in line and the amazing thing is the deeper the prey the higher they dive.
The graph shows the relationship between the horizontal distance that is the height from the land, vertical distance is the point from where its destination point is located, so at the starting point of the flight it has more distance from the ground means flying at a higher level , as the time passes it reaches to the closer point of its destination point so the altitude of the flight decreases with the decrease in the vertical distance and at a particular distance reaches its point of destination.

b. The decrease in the vertical distance of the pelican. is due to its flight to the destination point as it requires to stop flying to reach it, so in order to have a smooth landing on the ground , the bird gradually decreases its speed by decreasing its altitude.

Question 2.
A fully-charged battery loses its charge at a constant rate until it has no charge left. You plug it in, and it fully recharges at a constant rate. Then it loses its charge at a constant rate until it has no charge left. Sketch a graph that represents this situation.
Answer:  The graph is

Explanation:
In the graph , let the x-axis be time and y-axis be the battery charge ,
A fully-charged battery loses its charge at a constant rate until it has no charge left. So, line segment starts from 100 and decreases until it touches the x-axis.
You plug it in, and it fully recharges at a constant rate. Thus, line segment increases at a constant rate until it reaches 100
Then it loses its charge at a constant rate until it has no charge left. line segment decreases again at a constant rate until it again touches the x-axis .

Self-Assessment for Concepts & Skills
Solve each exercise. Then rate your understanding of the success criteria in your journal.

Question 3.
ANALYZING GRAPHS
The graph shows the growth rate of a plant over time.

a. Describe the change in growth rate.
b. Write an explanation for the decrease in growth rate and the increase in growth rate.
Answer: the answers are given below

Explanation:
a. the change in growth rate of a plant over the time is given by its size and height , So as the time passes the growth rate is constant from the the start and from a particular time the growth rate has been dropping slightly due to external or internal reasons of a plant and again at some time the growth rate is increasing at a constant rate until it reaches to its perfect growth of a plant.

b. The decrease in growth rate of the plant is due to some external causes like weather, rain, sunlight , watering, and the soil may effect its growth rate and the increase in growth rate is probably due to its soil fertility and sufficient sunlight providing sufficient chlorophyll.

Question 4.
SKETCHING GRAPHS
As you snowboard down a hill, you gain speed at a constant rate. You come to a steep section of the hill and gain speed at a greater constant rate. You then slow down at a constant rate until you come to a stop. Sketch a graph that represents this situation.
Answer: The graph is

Explanation:
In the graph , let the x-axis be time and y-axis be the speed ,
As you snowboard down a hill, you gain speed at a constant rate, line segment decreases at a constant rate
You come to a steep section of the hill and gain speed at a greater constant rate, line segment becomes steeper i.e., the line segment decreases at a high constant rate.
You then slow down at a constant rate until you come to a stop, line segment becomes flatter i.e., the constant rate of decrease becomes less until it touches its x-axis

Self-Assessment for Problem Solving
Solve each exercise. Then rate your understanding of the success criteria in your journal.

Question 5.
Two rowing teams are in a race. The graph shows their distances from the finish line over time. Describe the speed of each team throughout the race. Then determine which team finishes first.

Answer: Team B will finishes race first.

Explanation:
Team A , The relationship between the time and distance from the finish line is given in the graph,
At starting point Team A has maintained a fair speed at the Beginning of the race and has been a little slow while reaching out to the destination point, and for a while they have been balancing the speed with the distance representing a curving point in the graph and directly dropping to the finish line drastically creating a slope, until it reaches in the x-axis line.
Team B , The relationship between the time and distance from the finish line is given in the graph,
As same as the Team A , Team B has a perfect start but it has been a way different them Team A because Team B has a game plan to win the race, as shown in the graph they have maintained a constant speed while reaching out to the destination and also having a smooth drift at a level of decreasing their distance from the finish line.

Team B will  finishes the race first because they are having a constant and smooth decreasing speed which comes to an end gradually at the finishing line.

Question 6.
DIG DEEPER!
The graphs show the movements of two airplanes over time. Describe the movement of each airplane.

Answer: Detailed explanation is given below.

Explanation:
As shown in the graph , x-axis is time and y-axis be the height above ground
Airplane A, the line segment at a constant rate at the time of starting of the takeoff and then drops to a point while decreasing in the height to the ground for landing and for a constant time it is at the ground level until again it takes off  having an increase in the height from the ground level , at last it maintains a constant speed.

Airplane B, the line segment at a constant rate at the time of starting of the takeoff and then drops to a point while decreasing in the height to the ground for landing and for a constant time it is at the ground level until again it takes off  having an increase in the height from the ground level , at last it maintains a constant speed.
It is as same as the airplane A.

Analyzing and Sketching Graphs Homework & Practice 7.5

Review & Refresh

Does the table or equation represent a linear or nonlinear function? Explain.
Question 1.

Answer: y = -0.5x + 11.5 is a linear function.

Explanation:
Ordered pairs are (-5 , 14) , (-1 , 12) , (3 , 10) , (7 , 8)
Plot the points in the table , Draw a line through the points
First find the slope m of the line containing the two given points (3 , 10) and (7 , 8)
m = (y₂-y₁) / (x₂-x₁)
m= (8 – 10) / (7 – 3)
m = -2/4
m = -0.5
substitute the slope in the(3 , 10) to get point slope to form a line.
y-y₁ = m (x-x₁)
y – 10 = -0.5 ( x – 3)
y – 10 = -0.5x + 1.5
y = -0.5x + 1.5 + 10
y = -0.5x + 11.5
So , the linear equation is y = -0.5x + 11.5
And it is a linear function.

The graph is

Question 2.
y = x² + 8
Answer: The function is linear  when m= 1 and c = 8.

Explanation:
Given , y = x² + 8 ,
slope m = 1
c = 8
So, the function is linear when m = 1 and c= 8.

Graph the linear equation.
Question 3.
– 4x + y = – 1
Answer: The graph is

Explanation:
we can write – 4x + y = – 1 as y = 4x – 1
Given , y = 4x – 1  , we know y = mx + c , where m = slope , c = constant
To obtain the graph , we should have ordered pairs ,
So , if x = 0 , then y = 4(0) – 1 = -1 . co-ordinates are (0 , -1)
if x = 1 , then y = 4(1) – 1 = 3 . co-ordinates are (1 , 3)
if x = 2 , then y = 4(2) – 1 = 7 , co-ordinates are (2 , 7)
if x = 3 , then y = 4(3) – 1= 11 , co-ordinates are (3 , 11)
The co-ordinates (0 , -1) , (1 , 3) , (2 , 7) , (3 , 11) form a straight line .

Question 4.
2x – 3y = 12
Answer: The graph is

Explanation:
we can write  2x – 3y = 12 as y = \(\frac{2x-12}{3}\) or y = \(\frac{2}{3}\)x – 4
Given , y =\(\frac{2}{3}\)x – 4 , we know y = mx + c , where m = slope , c = constant
To obtain the graph , we should have ordered pairs ,
So , if x = 0 , then y = \(\frac{2}{3}\)0 – 4= – 4 . co-ordinates are (0 , -4)
if x = 1 , then y = \(\frac{2}{3}\)1 – 4 = 0.66 – 4 = -3.3 . co-ordinates are (1 , -3.3)
if x = 2 , then y = \(\frac{2}{3}\)2 – 4 =0.66(2) – 4 =1.3 – 4 = -2.6, co-ordinates are (2 , -2.6)
if x = 3 , then y = \(\frac{2}{3}\)3 – 4 = 0.66(3) – 4 = 1.98 – 4 = -2.0  , co-ordinates are (3 , -2.0)
The co-ordinates (0 , -4) , (1 , -3.3) , (2 , -2.6) , (3 , -2) form a straight line .

Question 5.
5x + 10y = 30
Answer: The graph is

Explanation:
5x + 10y = 30 can be written as y = -0.5x + 3
take 5 common on both sides we get
x + 2y = 6
y = \(\frac{-x + 6}{2}\)
y = \(\frac{-x}{2}\) + 6
y = -0.5x + 3
Given , y =-0.5x + 3 , we know y = mx + c , where m = slope , c = constant
To obtain the graph , we should have ordered pairs ,
So , if x = 0 , then y = -0.5(0) + 3 = 3 . co-ordinates are (0 , 3)
if x = 1 , then y = -0.5(1) + 3= 2.5 . co-ordinates are (1 , 2.5)
if x = 2 , then y = -0.5(2) + 3 = 4 , co-ordinates are (2 , 4)
if x = 3 , then y = -0.5(3) + 3 = 4.5  , co-ordinates are (3 , 4.5)
The co-ordinates (0 , 3) , (1 , 2.5) , (2 , 4) , (3 , 4.5)does not form a straight line .

Concepts, Skills, &Problem Solving

MATCHING DESCRIPTIONS WITH GRAPHS The graph shows your speed during a run. Match the verbal description with the part of the graph it describes. (See Exploration 1, p. 301.)

Question 6.
You run at a constant speed.
Answer: C

Explanation:
Because the line segment of the graph at point C show that the running speed is constant for a particular time ,
Thus forming a straight horizontal line.

Question 7.
You slow down at a constant rate.
Answer: D

Explanation:
Because the line segment of the graph at point D show that the running speed is decreasing at a constant rate for a particular time ,
Thus forming a straight steep line down the time axis.

Question 8.
You increase your speed at a constant rate.
Answer: A

Explanation:
Because the line segment of the graph at point A show that the running speed is increasing at a constant rate at a starting point of the race on time ,
Thus forming a slope in the graph.

Question 9.
You increase your speed at a faster and faster rate.
Answer: B

Explanation:
Because the line segment of the graph at point B show that the running speed is increasing at a faster rate after starting the race and maintaining a gradual growth of the speed and after reaching the next point speed is doubled from before ,
Thus forming a slope with a curve in the graph.

ANALYZING GRAPHS Describe the relationship between the two quantities.
Question 10.

Answer: As the Time passes there will be increase in the volume.

Explanation:
The graph shows the relation between the volume and time of a Balloon , To fill up the balloon with air, with the inlet of air increases in volume and the time taken to fill the balloon with air is shown ,
The line segment starts at initial point stating the balloon at a no air state, then gradually increases with constant halts having constant volume at a particular time and vice versa.

So, As the Time passes there will be increase in the volume.

Question 11.

Answer: As the times passes Dollars are maintaining imbalance.

Explanation:
The relationship between the time and dollars is given in the graph, As we all know money is never ever constant with time , As if it only increases or decreases or having both simultaneously , in this graph the line segment is having a steep and at some point of time it is maintaining a slight growth constantly with the time.

So, As the times passes Dollars are maintaining imbalance.

Question 12.

Answer: An engine power is directly proportional to the engine speed and its horse power

Explanation:
The relationship between the engine speed and horse power is given in the graph, Generally every automobile is is defined as the best for its horse power which is the heart of the engine and it highlights the speed of the vehicle, Here engine power is defined by the horse power and the engine speed the line segment is having a curve increment in the horse power due to the increase in engine speed.

So, An engine power is directly proportional to the engine speed and its horse power

Question 13.

Answer: As time increases the process of grams decaying will be faster.

Explanation:
The relationship between grams and time is given in the graph, its obvious that every product has its own expiry date, and if it crosses that its starts to decay, the graph implies that with the increase time the quality of the gram decreases or grams start to decay . The line segment in the graph shows that the gradually decrease indicating the spoiling rate of the grams  with rate of change of time.

So, As time increases the process of grams decaying will be faster.

Question 14.

Answer: At the particular intervals of time hair growth has stopped and at regular intervals again starts growing with respect to the time.

Explanation:
The graph shows the relationship between the length of the hair and time taken to the growth of the hair, of course hair growth is not constant every time, here we have the graph with the line segment  not constant and having breaks at the times of interval.

So, At the particular intervals of time hair growth has stopped and at regular intervals again starts growing with respect to the time.

Question 15.

Answer: In a period of time the balance will be reduced gradually at regular intervals maintaining a constant balance to clear the loan.

Explanation:
The relationship between the balance of the loan with the time period of the loan to be cleared, The loan should be cleared in the time limit and should maintain a neat balance, every increase in time period the balance is debited from the loan , there will be decrease in the balance and gaps are occurred in the graph.

so, In a period of time the balance will be reduced gradually at regular intervals maintaining a constant balance to clear the loan.

Question 16.
ANALYZING GRAPHS
Write an explanation for the relationship shown in the graph in Exercise 10.
Answer: The graph shows the relation between the volume and time of a Balloon , To fill up the balloon with air, with the inlet of air increases in volume and the time taken to fill the balloon with air is shown ,
The line segment starts at initial point stating the balloon at a no air state, then gradually increases with constant halts having constant volume at a particular time and vice versa.

Question 17.
MODELING REAL LIFE
The graph shows the natural gas usage for a house.

a. Describe the change in usage from January to March.
b. Describe the change in usage from March to May.
Answer: a. The change in usage from January to March, As shown in the graph the usage is at peaks in the month of January and started to decrease after that month and continuing to decrease in the month of March.
b.  The change in usage from March to May, As shown in the graph the usage is continuing to decrease in the month of March and started to increase again in the month of May and maintaining to use highly from the month of May.

Explanation:
a. The change in usage from January to March, As shown in the graph the usage is at peaks in the month of January and started to decrease after that month and continuing to decrease in the month of March.

b.  The change in usage from March to May, As shown in the graph the usage is continuing to decrease in the month of March and started to increase again in the month of May and maintaining to use highly from the month of May.

SKETCHING GRAPHS Sketch a graph that represents the situation.
Question 18.
The value of a television decreases at a constant rate, and then remains constant.
Answer: The graph is

Explanation:
Draw the axis and label the x- axis as time and y- axis as value, then sketch the graph,
The value of the television decreases at a constant rate: line segment starts to decrease at a constant rate,
And then remains constant, after reaching a certain value : line segment becomes parallel to horizontal axis.

Question 19.
The distance from the ground changes as your friend swings on a swing.
Answer: The graph is

Explanation:
Your friend starts close to the ground and then swings up. Then she falls back down close to the ground again and swings back . When she swings back, she gets higher than when she was swinging forward, she then starts to swing forward again getting close to the ground and then going up even higher than when she was swinging backward, she continues to getting higher and higher every time she swings forwards and backwards,

Question 20.
The value of a rare coin increases at a faster and faster rate.
Answer: The graph is

Explanation:
Draw the Axis and label them as x-axis as time and y – axis as distance,
The value of a rare coin increases at a faster and faster rate , so the curve moves upwards at an increasing rate.

Question 21.
You are typing at a constant rate. You pause to think about your next paragraph and then you resume typing at the same constant rate.
Answer: The graph is

Explanation:
A constant rate means that portion of the graph is linear , pausing means the number of words stays constant, typing again at the same constant rate means the last piece of the graph is linear again with the same slope as the first portion of the graph.

Question 22.
CRITICAL THINKING
The graph shows the speed of an object over time.

a. Sketch a graph that shows the distance traveled by the object over time.
b. Describe a possible situation represented by the graphs.
Answer: a. The distance and time are directly proportional to each other.
b. As time passes the speed and time are relatively balancing each other in the graph.

Explanation:
a. The graph is 
In this graph the relationship between distance and time is shown, for example , let the object be a bike, the time taken to reach the destination for the bike is directly proportional to the distance travelled , So as time passes the distance is gradually increasing from the starting point.

So, the distance and time are directly proportional to each other.

b. Th graph shown , is the relationship between the speed and the time , let the object moving be Train,
it is running between the station so it has to be halted in the stations to be listed in the stoppings , So the line segment is started with a constant speed with the time and again at the time interval dropping the speed with respect to time it has maintaining the same speed .

So, As time passes the speed and time are relatively balancing each other in the graph.

Question 23.
MODELING REAL LIFE
The graph shows the average scores of two bowlers from the start of a season to the end of the season.

a. Describe each bowler’s performance.
b. Who had a greater average score most of the season? Who had a greater average score at the end of the season?
c. Write an explanation for the change in each bowler’s average score throughout the bowling season.
Answer: All the answers are explained below

Explanation:
a.  Bowler A : As the graph represent the relationship between the score and the week, bowler A has started with the good take off and having able to grasp the same energy his performance on the goals are increasing rapidly over the week and had the good score than Bowler B.

Bowler B : As the graph represent the relationship between the score and the week, bowler B has started with the good take off but as the time passes the performance of the  bowler is has been intended to decrease his scores gradually over the week.

b.  Bowler A and Bowler B had a greater average score most of the season, but Bowler A had a greater average score at the end of the season

c. Bowler A has the same energy his performance on the goals are increasing rapidly over the week and had the good score than Bowler B. so it has a smaller change in average’s score in the bowling season .
While Bowler B has good take off but as the time passes the performance of the  bowler is has been intended to decrease his scores gradually over the week. so he has a drastic change in average’s score in the bowling season .

Question 24.
DIG DEEPER!
You can use a supply and demand model to understand how the price of a product changes in a market. The supply curve of a particular product represents the quantity suppliers will produce at various prices. The demand curve for the product represents the quantity consumers are willing to buy at various prices.

a. Describe and interpret each curve.
b. Which part of the graph represents a surplus? Explain your reasoning.
c. The curves intersect at the equilibrium point, which is where the quantity produced equals the quantity demanded. Suppose that demand for a product suddenly increases, causing the entire demand curve to shift to the right. What happens to the equilibrium point?
Answer:  All of them are explained below .

Explanation:
a. The supply curve of a particular product represents the quantity suppliers will produce at various prices, As shown in the graph, the relationship between the price and quantity is given , so if prices increases gradually Quantity increases .
The demand curve for the product represents the quantity consumers are willing to buy at various prices, As shown in the graph, the relationship between the price and quantity is given , so if prices decreases with increase in Quantity .

b. The graph does not implies any surplus because each demand and supply is given by their respective curve over the prices and quantity

c. As shown in the graph, The curves intersect at the equilibrium point, which is where the quantity produced equals the quantity demanded. Given, that demand for a product suddenly increases, causing the entire demand curve to shift to the right. Then the equilibrium point will be pointed where the two curves meet after the change in the demand graph so change in the supply graph is also possible.

Functions Connecting Concepts

Using the Problem-Solving Plan
Question 1.
The table shows the lengths x (in inches) and weights y(in pounds) of several infants born at a hospital. Determine whether weight is a function of length. Then estimate the weight of an infant that is 20 inches long.

Understand the problem.
You know the lengths and weights of several infants. You are asked to determine whether weight is a function of length and to estimate the weight of a 20-inch-long infant.

Make a plan.
Determine whether any of the lengths are paired with more than one weight. Then use a graphing calculator to find an equation that represents the data. Evaluate the equation when x = 20 to estimate the weight of a 20-inch-long infant.

Solve and check.
Use the plan to solve the problem. Then check your solution.
Answer: Weight is the function of the length

Explanation:
From the table we have , Each length has only one weight , so weight is a function of length.
First find the slope m of the line containing the two given points (19.3 , 7.3) and (18.9 , 6.5)
m = (y₂-y₁) / (x₂-x₁)
m= (6.5 – 7.3) / (18.9 – 19.3)
m = 0.2
substitute the slope in the (19.3 , 7.3) to get point slope to form a line.
y-y₁ = m (x-x₁)
y – 7.3 = 0.2 ( x – 19.3)
y – 7.3 = 0.2x – 3.86
y = 0.2x – 3.86 + 7.3
y = 0.2x + 3.4
So ,  y = 0.2x + 3.4 is linear function.

For x = 20 ,
y = 0.2 (20) + 3.4
y = 4 + 3.4
y = 7.4

So, The weight of an infant that is 20 inches long. is 7.4.

Question 2.
Each mapping diagram represents a linear function. At what point do the graphs of the functions intersect? Justify your answer.

Answer:  The point of intersection is (-1, -4)

Explanation:
Function 1 – Ordered pairs are ( -8 , 24 ) , ( -3 , 4 ) , ( -1 , -4 ) , ( 1 , -12) .
Function 2 – Ordered pairs are ( 6 , 17 ) , ( 10 , 29 ) , ( 13 , 38 ) , ( 15 , 44 ) .
Graph the points we get, So, The point of intersection is (-1,-4).

Performance Task

Heat Index
At the beginning of this chapter, you watched a STEAM Video called “Apparent Temperature.” You are now ready to complete the performance task related to this video, available at BigIdeasMath.com. Be sure to use the problem-solving plan as you work through the performance task.

Answer:

Functions Chapter Review

Review Vocabulary

Write the definition and give an example of each vocabulary term.

Input: Ordered pairs can be used to show inputs and outputs , inputs are represented by x

Output: Ordered pairs can be used to show inputs and outputs , Outputs are represented by y

Relation: A relation pairs inputs with outputs

Mapping diagram: A relation can be represented by ordered pairs or mapping diagrams.

Function: The relation that pairs each input with exactly one output is a function.

Function rule: it is an equation, that describes the relationship between inputs(independent variables) and outputs(dependent variables).

Linear function: A linear function is a function whose graph is a straight line i.e., non vertical line . A linear can be written in the form y = mx + c , where m is the slope and c is the y intercept

Non linear function: The graph of a linear function shows a constant rate of change, A non linear function does not have a constant rate of change, So its graph is a not a line.

Graphic Organizers
You can use an Example and Non-Example Chart to list examples and non-examples of a concept. Here is an Example and Non-Example Chart for functions.

Choose and complete a graphic organizer to help you study the concept.

1. linear functions
2. nonlinear functions
3. linear functions with positive slope
4. linear functions with negative slope

Answer: 1. linear functions

Chapter Self-Assessment

As you complete the exercises, use the scale below to rate your understanding of the success criteria in your journal.

7.1 Relations and Functions (pp. 275–280)
Learning Target: Understand the concept of a function.

List the ordered pairs shown in the mapping diagram. Then determine whether the relation is a function.
Question 1.

Answer: The ordered pairs are ( 1 , -4 ) , ( 3 , 6 ) , ( 5 , 0 ) , ( 7 , 6 ) , ( 7 , 8 ) and The relation is not a function .

Explanation:
As shown , The ordered pairs are ( 1 , -4 ) , ( 3 , 6 ) , ( 5 , 0 ) , ( 7 , 6 ) , ( 7 , 8 ) .
The input 7 has more than one output,
So, The relation is not a function .

Question 2.

Answer: ordered pairs are ( 0 , 0 ) , ( 1 , 10 ) , ( 2 , 5 ) , ( 3 , 15 ) and The relation is a function .

Explanation:
As shown , The ordered pairs are ( 0 , 0 ) , ( 1 , 10 ) , ( 2 , 5 ) , ( 3 , 15 ).
Each input has exactly one output ,
So, The relation is a function .

Question 3.

Answer: The ordered pairs are ( -1 , 0 ) , ( -1 , 1 ) , ( 0 , 1 ) , ( 1 , 2 ), ( 3 ,3 ) and The relation is not a function

Explanation:
As shown , The ordered pairs are ( -1 , 0 ) , ( -1 , 1 ) , ( 0 , 1 ) , ( 1 , 2 ), ( 3 ,3 ) .
The input -1 has more than one output ,
So, The relation is not a function .

Question 4.
For ordered pairs that represent relations, which coordinate represents the input? the output?

Answer: x coordinate is the input and y coordinate is the output

Explanation:
Ordered pairs from the given graph are ( 2 , 7 ) , ( 3 , 7 ) , ( 4 , 5 ) , ( 5 , 5 ) , ( 6 , 3 ) .
So , x coordinate is the input and y coordinate is the output

Question 5.
Draw a mapping diagram that represents the relation shown in the graph. Then determine whether the relation is a function. Explain.
Answer:

Explanation:
The mapping diagram is
each input has more than one output
So, relation is not a function.

Question 6.
The mapping diagram represents the lengths (in centimeters) of a rubber band when different amounts of force (in Newtons) are applied.

a. Is the length of a rubber band a function of the force applied to the rubber band?
b. Describe the relationship between the length of a rubber band and the force applied to the rubber band.
Answer:  a. Yes
b. For every increase in 0.7 in input there is an increment of 2 in output.

Explanation:
a. The ordered pairs are  ( 0 , 5 ) , ( 0.7 , 7 ) , ( 1.4 , 9 ) , ( 2.1 , 11 )
Each input has exactly one output
So, the length of a rubber band a function of the force applied to the rubber band.

b. For every increase in 0.7 in input there is an increment of 2 in output.

7.2 Representations of Functions (pp. 281–288)
Learning Target: Represent functions in a variety of ways.

Write a function rule for the statement.
Question 7.
The output is two less than the input.
Answer: y = x – 2

Explanation:
Let us say x is input and y is output , then
The output is two less than the input, will be
y = x – 2

Question 8.
The output is two more than one-fourth of the input.
Answer: y = \(\frac{x}{4}\) + 2

Explanation:
Let us say x is input and y is output , then
The output is two more than one-fourth of the input, will be
y = \(\frac{x}{4}\) + 2

Find the value of y for the given value of x.
Question 9.
y = 2x – 3; x = – 4
Answer: y = -8

Explanation:
Given, y = 2x
substitute x = -4 , we get
y = 2(-4)
y = -8.

Question 10.
y = 2 – 9x ; x = \(\frac{2}{3}\)
Answer: y = – 3.4

Explanation:
Given , y = 2 – 9x
substitute x = \(\frac{2}{3}\) , we get
y = 2 – 9 (0.6)
y = 2 – 5.4
y = – 3.4

Question 11.
y = \(\frac{x}{3}\) + 5; x = 6
Answer: y = 7.

Explanation:
Given, y = \(\frac{x}{3}\) + 5
substitute x = 6 , we get
y = \(\frac{6}{3}\) + 5
y = 2 + 5
y = 7.

Graph the function.
Question 12.
y = x + 3
Answer: The graph is

Explanation:
Given , y = x + 3  , we know y = mx + c , where m = slope , c = constant
To obtain the graph , we should have ordered pairs ,
So , if x = 0 , then y = 0 + 3 = 3 . co-ordinates are (0 , 4)
if x = 1 , then y = 1 + 3  = 4 . co-ordinates are (1 , 5)
if x = 2 , then y = 2 + 3 = 5 , co-ordinates are (2 , 6)
if x = 3 , then y = 3 + 3 = 6 , co-ordinates are (3 , 7)
The co-ordinates (0 , 3) , (1 , 4) , (2 , 5) , (3 , 6) form a straight line .

Question 13.
y = – 5x
Answer: The graph is

Explanation:
Given , y = – 5x  , we know y = mx + c , where m = slope , c = constant
To obtain the graph , we should have ordered pairs ,
So , if x = 0 , then y =- 5(0) = 0 . co-ordinates are (0 , 0)
if x = 1 , then y = – 5(1)  = – 5 . co-ordinates are (1 , – 5)
if x = 2 , then y = – 5(2) = -10 , co-ordinates are (2 , -10)
if x = 3 , then y =- 5(3) = -15 , co-ordinates are (3 , -15)
The co-ordinates (0 , 0) , (1 , -5) , (2 , -10) , (3 , -15) form a straight line .

Question 14.
y = 3 – 3x
Answer: The graph is

Explanation:
Given , y =3 – 3x , we know y = mx + c , where m = slope , c = constant
To obtain the graph , we should have ordered pairs ,
So , if x = 0 , then y = 3 – 3(0) = 3 . co-ordinates are (0 , 3)
if x = 1 , then y = 3 – 3(1)  = 0 . co-ordinates are (1 , 0)
if x = 2 , then y = 3 – 3(2) = – 3 , co-ordinates are (2 , – 3)
if x = 3 , then y =3 – 3(3) = – 6 , co-ordinates are (3 , – 6)
The co-ordinates (0 , 3) , (1 , 0) , (2 , – 3) , (3 , – 6) form a straight line .

Question 15.
An online music store sells songs for $0.90 each.
a. Write a function that you can use to find the cost of buying songs.
b. What is the cost of buying 5 songs?
Answer: a. C = 0.90s
b. $4.5

Explanation:
a. The total cost is equal to the cost of each song times the number of songs, if each song is $0.90,
Then the total cost C of s songs is C = 0.90s.

b. Substituting s= 5 in C = 0.90s we get,
C = 0.90(5) = 4.5.
So, cost of 5 songs is $4.5.

7.3 Linear Functions (pp. 289–294)
Learning Target: Use functions to model linear relationships.

Use the graph or table to write a linear function that relates y to x.
Question 16.

Answer: The linear function is y = \(\frac{1}{3}\)x + 3.

Explanation:
In order to write the function we have to write the ordered pairs of the graph ,
Ordered pairs are  (3 , 4) , (0 , 3 ) , (-3 , 2) , ( -6, 1 )
First find the slope m of the line containing the two given points (0 ,3) and (-3, 2)
m = (y₂-y₁) / (x₂-x₁)
m= (2 – 3) / (-3 – 0)
m = -1 / -3 .
m = 1/3 .
Because the line crosses the y axis at ( 0, 3 ) , The y intercept is 3.
So , the linear function is y = \(\frac{1}{3}\)x + 3.

Question 17.

Answer: The linear function is y = −(0)x -7.

Explanation:
In order to write the function we have to write the ordered pairs of the graph ,
Ordered pairs are  (-2 , -7) , (0 , -7 ) , (2 , -7) , ( 4 , -7 )
First find the slope m of the line containing the two given points (0 ,-7) and (2, -7)
m = (y₂-y₁) / (x₂-x₁)
m= (-7 – (-7)) / (2 – 0)
m = 0 .
Because the line crosses the y axis at ( 0, -7 ) , The y intercept is -7.
So , the linear function is y = −(0)x -7.

Question 18.
The table shows the age x (in weeks) of a puppy and its weight y (in pounds).

a. Write and graph a linear function that relates y to x.
b. Interpret the slope and the y-intercept.
c. After how many weeks will the puppy weigh 33 pounds?
Answer: a. y = \(\frac{3}{2}\)x + 3
b. 3 pounds
c. Age is 20 weeks

Explanation:
a. In order to write the function we have to write the ordered pairs of the graph ,
Ordered pairs are  (6 , 12) , (8 , 15 ) , (10 , 18) , ( 12 , 21 )
First find the slope m of the line containing the two given points ((6 ,12) and (8 , 15)
m = (y₂-y₁) / (x₂-x₁)
m= (15 – 12) / (8 – 6)
m = 3/2 .
substitute the slope in the (6 ,12) to get point slope to form a line.
y-y₁ = m (x-x₁)
y – 12 = 3/2 ( x – 6)
2(y – 12) = 3(x – 6)
2y – 24 = 3x – 18
2y = 3x – 18 + 24
2y  = 3x + 6
So ,  2y  = 3x + 6 or y = \(\frac{3}{2}\)x + 3 is linear function.

b. The slope measures the rate of change of weight due to change in weeks, Here the slope of 3/2 means that as one week passes, weight of the puppy increases by 3/2 pounds.
y intercept measures the weight of the puppy, when it was born which is 3 pounds in this case measured by c.

c. put y = 33,
33 = \(\frac3}{2}\)x + 3
30 = \(\frac{3}{2}\)x
30 × 2 = 3x
x = 60/3
x = 20.
So, Age is 20 weeks.

7.4 Comparing Linear and Nonlinear Functions (pp. 295–300)
Learning Target: Understand differences between linear and nonlinear functions.

Does the table represent a linear or nonlinear function? Explain.
Question 19.

Answer: y = 3x – 8 is linear function.

Explanation:
Ordered pairs are (3 , 1 ) , (6 , 10) , (9 , 19) , (12 , 28)
Plot the points in the table , Draw a line through the points
First find the slope m of the line containing the two given points (3 , 1 ) and (6 , 10)
m = (y₂-y₁) / (x₂-x₁)
m= (10 – 1) / (6– 3)
m = 9/3
m = 3
substitute the slope in the (3 , 1) to get point slope to form a line.
y-y₁ = m (x-x₁)
y – 1 = 3 ( x – 3)
y – 1 = 3x – 9
y = 3x – 9 + 1
y = 3x – 8
So ,  y = 3x – 8 is linear function.

Question 20.

Answer: y = -x + 4 is linear function.

Explanation:
Ordered pairs are (1 , 3 ) , (3 , 1) , (5 , 1) , (7 , 3)
Plot the points in the table , Draw a line through the points
First find the slope m of the line containing the two given points (1 , 3 ) and (3 , 1)
m = (y₂-y₁) / (x₂-x₁)
m= (1 – 3) / (3– 1)
m = -2/2
m = -1
substitute the slope in the (1 , 3) to get point slope to form a line.
y-y₁ = m (x-x₁)
y – 3 = -1 ( x – 1)
y – 3 = -x + 1
y = -x + 1 + 3
y = -x + 4
So ,  y = -x + 4 is linear function.

Question 21.
Does the graph represent a linear or nonlinear function? Explain.

Answer: The graph represent a non linear function.

Explanation:
As shown in the graph linear function represents a  straight line to which not happened here,
So , the graph is non linear function

Question 22.
Does the equation y = 2.3x represent a linear or nonlinear function? Explain.
Answer: y = 2.3x is a linear function.

Explanation:
Given , y = 2.3x  , we know y = mx + c , where m = slope , c = constant
To obtain the graph , we should have ordered pairs ,
So , if x = 0 , then y = 2.3(0) = 0 . co-ordinates are (0 , 0)
if x = 1 , then y = 2.3(1) = 2.3 . co-ordinates are (1 , 2.3)
if x = 2 , then y = 2.3(2) = 4.6 , co-ordinates are (2 , 4.6)
The co-ordinates (0 , 0) , (1 , 2.3) , (2 , 4.6) form a straight line .
Each x input has only one y output so it is a function .
And it  forms a straight line when graphed .
So, y = 2.3x is a linear function.

7.5 Analyzing and Sketching Graphs (pp. 301–306)
Learning Target: Use graphs of functions to describe relationships between quantities.

Question 23.
Describe the relationship between the two quantities in the graph.

Answer: At certain point of time, line segment is increased with respect to time and maintained a constant change and again dropped gradually with the increase in time.

Explanation:
The relationship between the graph is population and time ,
At certain point of time, line segment is increased with respect to time and maintained a constant change and again dropped gradually with the increase in time.
So, the city population is not constant at all the time.

Sketch a graph that represents the situation.
Question 24.
You climb a climbing wall. You climb halfway up the wall at a constant rate, then stop and take a break. You then climb to the top of the wall at a greater constant rate.
Answer: The graph is 

Explanation:
You start climbing a wall at a constant rate so the first portion of the graph needs to be linear with a positive slope, you then take a break which means your height is constant so the second part of the graph needs to be a horizontal line, you then start climbing again at a constant rate, so the last part of the graph needs to be linear with a positive slope.

Question 25.
The price of a stock increases at a constant rate for several months before the stock market crashes. The price then quickly decreases at a constant rate.
Answer: The graph is

Explanation:
The stock price is increasing at a constant rate so the first part of the graph needs to be linear with positive slope, Then price begins to drop quickly so the second part of the graph needs to be linear with a steep negative slope.

Question 26.
The graph shows the sales of two companies during a particular year.

a. Describe the sales of each company.
b. Which company has greater total sales for the year?
c. Give a possible explanation for the change in each company’s sales throughout the year.
Answer: All The explanation is given below

a. Company A – The sales of the company is increasing at a constant rate so the first part of the graph needs to be linear with positive slope, and decreasing with a slight negative steep and again increasing at a constant rate increasing the sales of the company

Company B – The sales of the company is increasing at a constant rate so the first part of the graph needs to be increase in curve with a slight decrease in the graph leading to decrease in sales and vise versa.

b. Company A has the greater total sales for the year compared to Company B, with maintaining the sales up to the mark without losses.

c. The change in each company’s sales throughout the year,  Company A – The sales of the company is increasing at a constant rate so the first part of the graph needs to be linear with positive slope,
Company B – The sales of the company is increasing at a constant rate so the first part of the graph needs to be increase in curve with a slight decrease in the graph leading to decrease in sales

Functions Practice Test

Question 1.
List the ordered pairs shown in the mapping diagram. Then determine whether the relation is a function.

Answer: The relation is a function

Explanation:
As shown , Ordered pairs are the combinations of input and output
So , Ordered pairs are ( 2 , 9 ) , ( 4 , 9 ) , ( 6 , 10 ) , ( 8 , 11 ) .
Each input has exactly one output ,
So , The relation is a function .

Question 2.
Draw a mapping diagram that represents the relation. Then determine whether the relation is a function. Explain.

Answer: The mapping diagram is

Explanation:
Ordered pairs from the given graph are ( -3 , 5 ) , ( -1 , 1 ) , ( -1 , 3 ) , ( 1 , 2 ) , ( 3 , 4 ) .
Each input has exactly one output ,
So , The relation is a function .

Question 3.
Write a function rule for “The output is twice the input.”
Answer: y = 2x

Explanation:
Let us say x is input and y is output , then
The output is twice the input. will be
y = 2x

Question 4.
Graph the function y = 1 – 3x.
Answer: The graph is

Explanation:
Given , y = 1 – 3x  , we know y = mx + c , where m = slope , c = constant
To obtain the graph , we should have ordered pairs ,
So , if x = 0 , then y =1 – 3(0) = 1 . co-ordinates are (0 , 1)
if x = 1 , then y = 1 – 3(1)  = -2 . co-ordinates are (1 , -2)
if x = 2 , then y = 1 – 3(2) = -5 , co-ordinates are (2 , -5)
if x = 3 , then y =1 – 3(3) = -8 , co-ordinates are (3 , -8)
The co-ordinates (0 , 1) , (1 , -2) , (2 , -5) , (3 , -8) form a straight line .

Question 5.
Use the graph to write a linear function that relates y to x.

Answer: The linear function is y = 0.5x – 1

Explanation:
In order to write the function we have to write the ordered pairs of the graph ,
Ordered pairs are  (-4 , -3) , (-2 , -2 ) , (0 , -1) , ( 2 , 0 )
First find the slope m of the line containing the two given points (0 , -1) and ( 2 , 0 )
m = (y₂-y₁) / (x₂-x₁)
m= (0 – (-1)) / (2 – 0)
m = 1 / 2 .
m = 0.5 .
Because the line crosses the y axis at ( 0, -1 ) , The y intercept is -1.
So , the linear function is y = 0.5x – 1 .

Question 6.
Does the table represent a linear or nonlinear function? Explain.

Answer: The linear function is y = −4x + 8

Explanation:
In order to write the function we have to write the ordered pairs of the graph ,
Ordered pairs are  (0 , 8) , (2 , 0 ) , (4 , -8) , ( 6 , -16 )
First find the slope m of the line containing the two given points (0 , 8) and (2 , 0 )
m = (y₂-y₁) / (x₂-x₁)
m= (0 – 8) / (2 – 0)
m = -4
Because the line crosses the y axis at ( 0, 8 ) , The y intercept is 8.
So , the linear function is y = −4x + 8.

Question 7.
The table shows the number of y meters a water-skier travels in x minutes.

a. Write a function that relates y to x.
b. Graph the linear function.
c. At this rate, how many kilometers will the water-skier travel in 12 minutes?
d. Another water-skier travels at the same rate but starts a minute after the first water-skier. Will this water-skier catch up to the first water-skier? Explain.
Answer: All the answers are given below

Explanation:
Ordered pairs are  (1 , 600) , (2 , 1200 ) , (3 , 1800) , ( 4 , 2400 ) , (5 , 3000)
First find the slope m of the line containing the two given points(1 , 600) and (2 , 1200 )
m = (y₂-y₁) / (x₂-x₁)
m= (1200 – 600) / (2 – 1)
m = 600
So, the line is of the form y = 600x + c
put x= 3 and y = 1800 in the above equation we get,
1800 = 600(3) + c
c = 1800 – 1800
c = 0.
So, The line is y = 600x.

b. The graph is

c. put x = 12 in y = 600x
y = 600(12)
y = 7200
7200 meters, i.e., 7.2km

d. Another water skier travels at the same rate but starts a minute after the first water skier, Since both are travelling at the same rate , the water skier who was late will always be behind the first water skier.

Question 8.
The graph shows the prices of two stocks during one day.

a. Describe the changes in the price of each stock.
b. Which stock has a greater price at the end of the day?
c. Give a possible explanation for the change in the price of Stock B throughout the day.
Answer: Detailed Explanation is given below.

Explanation:
a. The changes in the price of each stock is Stock A has the constant increase in stock for a particular time and maintains a constant price forming a straight line in the graph, and again decreasing with a negative slope and vise versa, while Stock B is having steep negative slope that is decreasing in the prices with the time and having a constant horizontal line. and having a positive increase in the slope and same repeats again.

b. stock B has a greater price at the end of the day, having a positive increase in the slope

c. The change in the price of Stock B throughout the day, is having steep negative slope that is decreasing in the prices with the time and having a constant horizontal line. and having a positive increase in the slope and same repeats again, compared to stock A .

Question 9.
You are competing in a footrace. You begin the race by increasing your speed at a constant rate. You then run at a constant speed until you get a cramp and have to stop. You wait until your cramp goes away before you start increasing your speed again at a constant rate. Sketch a graph that represents the situation.
Answer: The graph is

Explanation:
You begin the race by increasing your speed at a constant rate so the first portion of the graph needs to be linear with a positive slope , you then run at a constant speed so the next portion of the graph needs to be horizontal line , you then stop and take a break , so your speed is zero, which means the next portion of the line needs to be
horizontal line on the x axis , you then increase your speed again at a constant rate sop that the last portion of the graph needs to be linear with a positive slope

Functions Cumulative Practice

Question 1.
What is the slope of the line?

Answer: Not in the options but the answer is m = -4/3

Explanation:
Ordered pairs are  (-4 , 5) , (1 , -3 ),
First find the slope m of the line containing the two given points
m = (y₂-y₁) / (x₂-x₁)
m= (-3 – 5) / (2 – (-4))
m = -8/6
m = -4/3.

Question 2.
Which value of a makes the equation 24 = \(\frac{a}{3}\) – 9 true?
F. 5
G. 11
H. 45
I. 99
Answer: I. 99

Explanation:
Substitute a = 99 , in the given equation we get,
24 = \(\frac{a}{3}\) – 9
24 = \(\frac{99}{3}\) – 9
24 = 33 – 9
24 = 24.
So, last option is the correct answer.

Question 3.
A mapping diagram is shown.

What number belongs in the box so that the equation describes the function represented by the mapping diagram?

Answer: m = 7 , y = 7x + 5

Explanation:
Ordered pairs are  (4 , 33) , (7 , 54 ), (10 , 75) , (13 , 96 ),
First find the slope m of the line containing the two given points (4 , 33) and (7 , 54 )
m = (y₂-y₁) / (x₂-x₁)
m= (54 – 33) / (7 – 4)
m = 21/3
m = 7.
So, y = 7x + 5

Question 4.
What is the solution of the system of linear equations?
3x + 2y = 5
x = y + 5
A. (3, – 2)
B. (- 2, 3)
C. (- 1, 4)
D. (1, – 4)
Answer: A. (3, – 2)

Explanation:
Given 3x + 2y = 5
Then substitute , x = y + 5 in the above equation
3( y + 5) + 2y = 5
3y + 15 + 2y = 5
5y + 15 = 5
5( y + 3) = 5
y + 3 = 1
y = 1 – 3
y = -2,
substitute y = -2 in x = y + 5 then
x = 3
So, (3 , -2)

Question 5.
The director of a research lab wants to present data to donors. The data show how the lab uses a large amount of donated money for research and only a small amount of money for other expenses. Which type of display best represents these data?
F. box-and-whisker plot
G. circle graph
H. line graph
I. scatter plot
Answer: I. scatter plot

Explanation:
Scatter plot is the best graph for this type of data where vertical axis will show the amount of money and Horizontal axis will show research and other expenses.

Question 6.
Which graph shows a nonlinear function?

Answer: option B

Explanation:
As all the other options are representing the linear function that is forming a straight line  expect for option B , it is representing a non linear equation.

Question 7.
Which equation of a line passes through the point (—2, 3) and has a slope of \(\frac{3}{4}\)?

Answer:  F. y – 3 = \(\frac{3}{4}\)(x + 2)

Explanation:
Given, y – 3 = \(\frac{3}{4}\)(x + 2)
it is in the form of y = mx + c
so, slope m = \(\frac{3}{4}\)
Substitute the given points in this equation that is x = -2 and y = 3
3 – 3 = \(\frac{3}{4}\)(-2 + 2)
0 = 0.
So, F is the correct option.

Question 8.
The tables show the sales (in millions of dollars) for two companies over a five-year period.

Part A Does the first table show a linear function? Explain your reasoning.
Part B Does the second table show a linear function? Explain your reasoning.
Answer: Part A, the first table shows a linear function,
Part B the second table shows a linear function.

Explanation:
Part A – ordered pairs are (1 , 2) , (2 , 4) , (3 , 6) , (4 , 8) , (5 , 10)
Each input has exactly one output and forms a straight line when graphed
So, it is a linear function.

Part B – ordered pairs are (1 , 1) , (2 , 1) , (3 , 2) , (4 , 3) , (5 , 5)
Each input has exactly one output and does not form a straight line when graphed
So, it is a linear function.

Question 9.
The equations y = – x + 4 and y = \(\frac{1}{2}\)x – 8 form a system of linear equations. The table shows the values of y for given values of x.

What can you conclude from the table?
A. The system has one solution, when x = 0.
B. The system has one solution, when x = 4.
C. The system has one solution, when x = 8.
D. The system has no solution.
Answer: C. The system has one solution, when x = 8.

Explanation:
Given , y = – x + 4 and y = \(\frac{1}{2}\)x – 8
for x = 8 we have
y = -8 + 4 = -4
y = 0.5(8) – 8 = 4 – 8 = -4
Both the equations have one solution for x = 8
So, The system has one solution, when x = 8.

Question 10.
The vertices of a triangle are A (- 1, 3), B (1, 2), and C (- 1, – 1). Dilate the triangle using a scale factor of 2. What is the y-coordinate of the image of B?

Answer: The New right angle triangle is larger than the original one So , its a increase .

Explanation:
Given , (- 1, 3),  ( 1, 2 ),  (- 1, -1) these pairs form a right angle triangle
K = 2 , For the dilation figure multiply the 3 with the given ordered pairs , then
(- 1, 3) × 2 = ( -2 , 6)
( 1, 2) × 2 = ( 2 , 4)
(- 1, -1) × 2 = (-2 , -2)
From these new ordered pairs we form a new  right angle triangle

The New right angle triangle is larger than the original one So, it’s an increase.

Conclusion

Enhance your problem-solving skills and overall math proficiency using the Big Ideas Math Grade 8 Chapter 7 Functions. Give your best in exams by solving the Problems Over here aligned as per the Latest BIM Textbooks. In case of any suggestions do leave us your queries via the comment box so that we can get back to you.

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Big Ideas Math 1st Grade Answer Key Chapter 10 Measure and Compare Lengths

Here we can discuss different topics on Measuring and Comparing, Order Objects by Length, Compare Lengths Indirectly, Measure Lengths, Measure More Lengths, Solve Compare Problems Involving Length, and so on. Compare the topic you learned here with the real time and understand the topic deeply. Those topics were being set up by the mathematical experts as indicated by the latest edition. Scroll down this page to get the answers to all the inquiries. Click on the links to look at the subjects shrouded in this chapter Measuring and comparing lengths.

Lesson 1 Order Objects by Length

• Lesson 10.1 Order Objects by Length
• Order Objects by Length Practice 10.1

Lesson 2 Compare Lengths Indirectly

• Lesson 10.2 Compare Lengths Indirectly
• Compare Lengths Indirectly Practice 10.2

Lesson 3 Measure Lengths

• Lesson 10.3 Measure Lengths
• Measure Lengths Practice 10.3

Lesson 4 Measure More Lengths

• Lesson 10.4 Measure More Lengths
• Measure More Lengths Practice 10.4

Lesson 5 Solve Compare Problems Involving Length

• Lesson 10.5 Solve Compare Problems Involving Length
• Solve Compare Problems Involving Length Practice 10.5

Performance Task

• Measure and Compare Lengths Performance Task
• Measure and Compare Lengths Chapter Practice

Measure and Compare Lengths Vocabulary

Organize It

Review Words:
longer
shorter

Use the review words to complete the graphic organizer.

Answer:
The first image longer and the second image is shorter.

Explanation:

Define It

Use your vocabulary cards to identify the words. Find each word in the word search.

Answer:

Lesson 10.1 Order Objects by Length

Explore and Grow

Draw an object that is shorter than the pencil and longer than the crayon.

Answer:

Chalk is smaller than crayon.

Show and Grow

Question 1.
Order from longest to shortest.

_____________, _____________, _____________
Answer:
Purple, Pink, Blue.

Explanation:
In the above image, the purple color tube is the longest after that the pink color tube is the longest, and the blue color tube is the shortest.

Question 2.
Order from shortest to longest.

_____________, _____________, _____________
Answer:
Yellow, Green, Black.

Explanation:
In the above image, the yellow brush is the longest, and then the green brush is the longest. The black brush is the shortest.

Apply and Grow: Practice

Question 3.
Order from longest to shortest.

_____________, _____________, _____________
Answer:
Purple crayon, Red crayon, Green crayon.

Explanation:
In the above image, the Purple crayon is the longest, and then the red crayon is the longest. The green crayon is the shortest.

Question 4.
Order from shortest to longest.

_____________, _____________, _____________
Answer:
The order from shortest to longest is, Pink is the shortest, green is the largest, and blue is the longest.

Explanation:
In the above image, we can see that the order from shortest to longest is, Pink is the shortest, green is the largest, and blue is the longest.

Question 5.
YOU BE THE TEACHER
Your friend ordered from shortest to longest. Is your friend correct? Explain.

  Yellow   ,          green  ,           red   
Answer:
No, my friend is not correct.

Explanation:
No, my friend is not correct. He represented the order from longest to shortest. In the above image, the red chill is the shortest, and the green chili is the longest. The black brush is the shortest.

Think and Grow: Modeling Real Life

Your yarn is longer than Newton’s. Descartes’s is longer than Newton’s and shorter than yours. Who has the longest yarn?

Draw a picture:

Who has the longest yarn?
You             Newton      Descartes
Answer:
My yarn is the longest yarn.

Explanation:

As my yarn is longer than Newton’s and Descartes’s is longer than Newton’s and shorter than mine. So the longest yarn is mine, and the largest yarn is Descartes. The shortest yarn is Newton’s.

Show and Grow

Question 6.
Descartes’s pencil is shorter than Newton’s. Yours is shorter than Newton’s and longer than Descartes’s. Who has the shortest pencil?
Draw a picture:

Who has the shortest pencil?
Descartes        Newton           You
Answer: Your pencil is shorted among the three.

Order Objects by Length Practice 10.1

Order from longest to shortest.

Question 1.

_____________, _____________, _____________
Answer:
The order of the bats from longest to the shortest is
Bat 2, Bat 1, Bat 3.

Explanation:
In the above image, we can see the longest bat is bat 2 and then the largest bat is bat 1. The shortest bat is bat 3.

Question 2.

_____________, _____________, _____________
Answer:
The order of the colors from longest to shortest is
Gold, Blue, Red.

Explanation:
In the above image, we can see that the longest color is gold and then the largest color is blue. The shortest color is red.

Question 3.
Order from shortest to longest.

_____________, _____________, _____________
Answer:
The order from shortest to longest is
Vine 3, Vine 1, Vine 2.

Explanation:
In the above image, we can see that the shortest vine is vine 3 after that the largest vine is vine 1 and the longest vine is Vine 2.

Question 4.
DIG DEEPER!
Use the clues to match. The red pencil is longer than the yellow pencil. The shortest pencil is blue.

Answer:
The order from shortest to longest is
Blue pencil, Yellow pencil, Red pencil.

Explanation:

As by the given clue, the Blue pencil is the shortest pencil, the Yellow pencil is the largest pencil, and the Red pencil is the longest pencil.

Question 5.
Modeling Real Life
Your jump rope is longer than Newton’s. Descartes’s is longer than Newton’s and shorter than yours. Who has the longest jump rope?

Who has the longest jump rope?
You           Newton        Descartes
Answer:
My rope is the longest jump rope.

Explanation:
As my jump rope is longer than Newton’s and Descartes’s jump rope longer than Newton’s and Newton’s jump rope is shorter than yours. So my jump rope is Longer, Newton’s jump rope is the Largest, and the shortest rope is Descartes.

Review & Refresh

Compare.

Question 6.
25 ○ 52
Answer:
25 > 52.

Explanation:
Given that to compare 25 and 52. So we can see that 52 is greater than 25.

Question 7.
41 ○ 44
Answer:
41 > 44.

Explanation:
Given that to compare 41 and 44. So we can see that 44 is greater than 41.

Lesson 10.2 Compare Lengths Indirectly

Explore and Grow

Use string to compare the keys. Which key is longer?
How do you know?

Answer:
Key 1 is longer than Key 2.

Explanation:
In the above image, we can see that key 1 is longer than key 2.

Show and Grow

Question 1.
Circle the longer object.

Answer:
The Pen is longer than Eraser.

Explanation:

In the above image, we can see that the Pen is longer than the Easer.

Question 2.
Draw a line through the shorter object.

Answer:

Explanation:

Apply and Grow: Practice

Question 3.
Draw a line through the shorter object.

Answer:
Object 2 is shorter than object 1.

Explanation:

In the above image, we can see that object two is shorter than object 1.

Question 4.
Circle the longer object.

Answer:
The spoon is longer than the brush.

Explanation:

We can see in the above image, that the spoon is longer than the brush. So we will round off the spoon.

Question 5.
DIG DEEPER
Which object is longer? Explain.

Answer:
The book’s shelf is longer than the key.

Explanation:

As we can see in the above image, that the books shelf is longer than the key. So we will circle the books shelf.

Think and Grow: Modeling Real Life

A green crayon is shorter than a blue crayon. The blue crayon is shorter than a yellow crayon. Is the green crayon longer than or shorter than the yellow crayon?

Draw a picture:

Longer              Shorter
Answer:
The green crayon is shorter than the yellow crayon.

Explanation:

Given that green crayon is shorter than blue crayon, and blue crayon is shorter than yellow crayon. So green crayon is shorter than yellow crayon.

Show and Grow

Question 6.
A yellow ribbon is longer than a pink ribbon. The pink ribbon is longer than a blue ribbon. Is the yellow ribbon longer than or shorter than the blue ribbon?

Draw a picture:

Longer            Shorter
Answer:
The Yellow ribbon is longer than the blue ribbon.

Explanation:

Given that yellow ribbon is longer than pink ribbon, and the pink ribbon is longer than a blue ribbon. So yellow ribbon is longer than the blue ribbon.

Compare Lengths Indirectly Practice 10.2

Question 1.
Circle the longer object.

Answer:
Object 1 is longer than object 2.

Explanation:
In the above image, we can see that object one is longer than image two. So we will circle object one.

Question 2.
Draw a line through the shorter object.

Answer:
Object two is shorter than object one.

Explanation:

In the above image, we can see that object two is shorter than object one. So we will circle object two.

Question 3.
DIG DEEPER
Use the clues to match.
The blue string is longer than the orange string.
The purple string is shorter than the orange string.

Answer:
The order of the strings from longest to shortest is
Blue string, Orange string, Purple string.

Explanation:

Given that the blue string is longer than the orange string, and the purple string is shorter than the orange string.
So the order of the strings is the blue string is longer, the orange string is the largest and the purple string is the shortest.

Question 4.
Modeling Real Life
A kayak is shorter than a canoe. The canoe is shorter than a paddle board. Is the kayak longer than or shorter than the paddle board?

Longer             Shorter
Answer:
The Kayak is shorter than the paddle board.

Explanation:
Given that the Kayak is shorter than a canoe, and the canoe is shorter than a paddle board. So the Kayak is shorter than the paddle board.

Review & Refresh

Question 5.
Circle the objects that have capacity as an attribute.

Answer:
The objects which have the capacity to store are circled in the below image.

Explanation:

The circled items have the capacity to store. As we can see that the lid jar is used to store any items, and chalk box is used to store chalk pieces, and glass is used to store water or any liquids, and a glue bottle is used to store glue. So these items have the capacity to store. And the remaining things do not have that much capacity to store.

Lesson 10.3 Measure Lengths

Explore and Grow

Find and measure the objects shown in your classroom.

Answer:
The length of the table is about two colored red tiles.

Explanation:
The table is measured by using the colored red tile. As each tile is equal to one unit. So the length of the table is two colored red tiles, which means two units. We must take each tile without gaps or any overlaps between them.

Answer:
The length of the pencil is about one tile.

Explanation:
The pencil is measured by using the colored red tile. As each tile is equal to one unit. So the length of the pencil is one colored red tile, which means one unit. We must take each tile without gaps or any overlaps between them.

Show and Grow

Measure

Question 1.

about ___________ color tile
Answer:
The length of the object is about one tile.

Explanation:
The object is measured by using the colored tile. As each tile is equal to one unit. So the length of the object is about one colored tile, which means one unit. We must take each tile without gaps or any overlaps between them.

Question 2.

about ___________ color tile

Answer:
The length of the object is about five colored tiles.

Explanation:
The object is measured by using the colored tile. As each tile is equal to one unit. So the length of the object is about five colored tiles, which means five units. We must take each tile without gaps or any overlaps between them.

Apply and Grow: Practice

Measure

Question 3.

about ___________ color tile
Answer:
The length of the object is about four colored tiles.

Explanation:
The object is measured by using the colored tile. As each tile is equal to one unit. So the length of the object is about four colored tiles, which means four units. We must take each tile without gaps or any overlaps between them.

Question 4.

about ___________ color tile
Answer:
The length of the object is about three colored tiles.

Explanation:
The object is measured by using the colored tile. As each tile is equal to one unit. So the length of the object is about three colored tiles, which means three units. We must take each tile without gaps or any overlaps between them.

Question 5.
MP Precision
Which picture shows the correct way to measure the straw?

Answer:
The red color tiles are the correct way to measure the straw.

Explanation:
In the above image, we can see that the correct way to measure the straw is the red color tiles.

Think and Grow: Modeling Real Life

Will the scissors fit inside a pencil case that is 7 color tiles long?

Circle:      Yes     No
Tell how you know:
Answer:
Yes, the scissors can fit inside a pencil case that is 7 color tiles long.

Explanation:
Yes, the scissors can fit inside a pencil case. As the length of the scissors is four colored tiles. As the scissors are measured by using the colored tile. And each tile is equal to one unit. So the length of the scissors is about four colored tiles, which means four units. And the pencil case is 7 colored tiles long. So the scissors can fit. We must take each tile without gaps or any overlaps between them.

Show and Grow

Question 6.
Will the cell phone fit inside a case that is 5 color tiles long?

Circle:     Yes     No
Tell how you know:
Answer:
No, the cell phone cannot fit inside a case of 5 color tiles long.

Explanation:
No, the cell phone cannot fit inside a case of 5 color tiles long. As the length of the cell phone is six colored tiles. And the cell phone is measured by using the colored tile. And each tile is equal to one unit. So the length of the cell phone is about six colored tiles, which means six units. And cannot fit inside 5 colored tiles long. We must take each tile without gaps or any overlaps between them.

Measure Lengths Practice 10.3

Measure.

Question 1.

about ___________ color tile
Answer:
The length of the object is about four colored tiles.

Explanation:
The object is measured by using the colored tile. As each tile is equal to one unit. So the length of the object is about four colored tiles, which means four units. We must take each tile without gaps or any overlaps between them.

Question 2.

about ___________ color tile
Answer:
The length of the object is about two colored tiles.

Explanation:
The object is measured by using the colored tile. As each tile is equal to one unit. So the length of the object is about two colored tiles, which means two units. We must take each tile without gaps or any overlaps between them.

Question 3.

about ___________ color tile
Answer:
The length of the glue stick is about three colored tiles.

Explanation:
The glue stick is measured by using the colored tile. As each tile is equal to one unit. So the length of the glue stick is about three colored tiles, which means three units. We must take each tile without gaps or any overlaps between them.

Question 4.
MP Reasoning
The green yarn is about 3 color tiles long. How long is the blue yarn?

about ___________ color tile
Answer:
The length of the blue yarn is about six color tiles.

Explanation:
Given that the length of the green yarn is three color tiles long, so we should find the length of the blue yarn. To measure the length of the blue yarn, we will take the colored tiles without gaps or any overlaps between them. As each tile is equal to one unit. So the length of the blue yarn is about six colored tiles, which means six units.

Question 5.
Modeling Real Life
Will the gift card fit inside an envelope that is 8 color tiles long?

Circle:      Yes      No
Tell how you know:
Answer:
Yes, the gift can fit inside an envelope that is 8 color tiles long.

Explanation:
Yes, the gif card can fit inside an envelope that is 8 color tiles long. As the length of the gift card is four colored tiles. And the gift card is measured by using the colored tile. And each tile is equal to one unit. So the length of the gift is about four colored tiles, which means four units. And the envelope is 8 colored tiles long. So the gift card can fit. We must take each tile without gaps or any overlaps between them.

Review & Refresh

Question 6.
Complete the fact family.
7 + 3 = _________                                     __________ – 3 = 7
_________ + _________ = _________             _________ – 7 = _________
Answer:
7 + 3 is 10,
3 + 7 is 10
10 – 3 is 7,
10 –  7 is 3.

Explanation:
As a fact family represents a group of math facts, equations, which are created by using the same set of digits. This fact family defines the relation between the three numbers which are involved. In addition or subtraction in a fact family, there will be four addition and subtractions will be created using these three numbers. So
7 + 3 is 10,
3 + 7 is 10
10 – 3 is 7,
10 –  7 is 3.

Lesson 10.4 Measure More Lengths

Explore and Grow

Find and measure the objects shown in your classroom two ways. What do you notice?

Answer:
The length of the table is about two colored red tiles and the length of the table which is measured using a paper clip is one paper clip.

Explanation:
The table is measured by using the colored red tile. As each tile is equal to one unit. So the length of the table is two colored red tiles, which means two units. We must take each tile without gaps or any overlaps between them. And to measure the table with the paper clip, we will place the paper clip without gaps or overlaps between them. And each paper clip is equal to about one unit. So the length of the table using the paper clip is one paper clip, which means one unit. We have noticed that if we take a colored tile, then the length of the table will be two color tiles. And if we take a paper clip then the length of the table will be one paper clip. So the length is doubled while measuring with color tile than the paper clip.

Answer:
The length of the pencil is about two colored red tiles and the length of the pencil which is measured using a paper clip is one paper clip.

Explanation:
The pencil is measured by using the colored red tile. As each tile is equal to one unit. So the length of the table is two colored red tiles, which means two units. We must take each tile without gaps or any overlaps between them. And to measure the table with the paper clip, we will place the paper clip without gaps or overlaps between them. And each paper clip is equal to about one unit. So the length of the pencil using the paper clip is one paper clip, which means one unit. We have noticed that if we take a colored tile, then the length of the table will be two color tiles. And if we take a paper clip then the length of the pencil will be one paper clip. So the length is doubled while measuring with color tile than the paper clip.

Show and Grow

Measure

Question 1.

about __________ color tiles about __________ paper clips
Answer:
The length of the brush is about four colored tiles and the length of the brush which is measured using a paper clip is two paper clips.

Explanation:
The brush is measured by using the colored tile. As each tile is equal to one unit. So the length of the table is two colored tiles, which means two units. We must take each tile without gaps or any overlaps between them. And to measure the brush with the paper clip, we will place the paper clip without gaps or overlaps between them. And each paper clip is equal to about one unit. So the length of the brush using the paper clip is one paper clip, which means one unit. We have noticed that if we take a colored tile, then the length of the table will be two color tiles. And if we take a paper clip then the length of the brush will be one paper clip. So the length is doubled while measuring with color tile than the paper clip.

Question 2.

about __________ color tiles about __________ paper clips
Answer:
The length of the skating board is about two colored tiles and the length of the pencil which is measured using a paper clip is one paper clip.

Explanation:
The skating board is measured by using the colored tile. As each tile is equal to one unit. So the length of the table is two colored tiles, which means two units. We must take each tile without gaps or any overlaps between them. And to measure the skating board with the paper clip, we will place the paper clip without gaps or overlaps between them. And each paper clip is equal to about one unit. So the length of the skating board using the paper clip is one paper clip, which means one unit. We have noticed that if we take a colored tile, then the length of the skating board will be two color tiles. And if we take a paper clip then the length of the pencil will be one paper clip. So the length is doubled while measuring with color tile than the paper clip.

Apply and Grow: Practice

Measure.

Question 3.

about __________ color tiles about __________ paper clips
Answer:
The length of the object is about six colored tiles and the length of the pencil which is measured using a paper clip is three paper clips.

Explanation:
The object is measured by using the colored tile. As each tile is equal to one unit. So the length of the object is six colored tiles, which means six units. We must take each tile without gaps or any overlaps between them. And to measure the table with the paper clip, we will place the paper clip without gaps or overlaps between them. And each paper clip is equal to about one unit. So the length of the object using the paper clip is three paper clips, which means three units. We have noticed that if we take a colored tile, then the length of the object will be six color tiles. And if we take a paper clip then the length of the pencil will be three paper clips. So the length is doubled while measuring with color tile than the paper clip.

Question 4.

about __________ color tiles about __________ paper clips
Answer:

The length of the sunscreen lotion bottle is about four colored tiles and the length of the sunscreen lotion bottle which is measured using a paper clip is two paper clips.

Explanation:
The sunscreen lotion bottle is measured by using the colored tile. As each tile is equal to one unit. So the length of the sunscreen lotion bottle is four colored tiles, which means four units. We must take each tile without gaps or any overlaps between them. And to measure the sunscreen lotion bottle with the paper clip, we will place the paper clip without gaps or overlaps between them. And each paper clip is equal to about one unit. So the length of the sunscreen lotion bottle using the paper clip is two paper clips, which means two units. We have noticed that if we take a colored tile, then the length of the sunscreen lotion bottle will be four color tiles. And if we take a paper clip then the length of the sunscreen lotion bottle will be two paper clips. So the length is doubled while measuring with color tile than the paper clip.

Question 5.
YOU BE THE TEACHER
Your friend says the pencil is more paper clips long than color tiles. Is your friend correct? Explain.

Answer:
No, my friend is not correct.

Explanation:
No, my friend is not correct. As we can see from the above problems that while measuring with the paper clip we need fewer paper clips than the color tiles. And the length is doubled while measuring with color tile than the paper clip. So my friend is not correct.

Think and Grow: Modeling Real Life

Your guitar is 33 color tiles long. Is your guitar more than or less than 33 paper clips long?

Circle:      more than 33         less than 33
Tell how you know:
Answer:
My guitar will be less than 33 paper clips.

Explanation:
My guitar will be less than 33 paper clips. As we can see from the above problems that while measuring with the paper clip we need fewer paper clips than the color tiles. And the length is doubled while measuring with color tile than the paper clip. So there will be fewer paper clips than the color tiles.

Show and Grow

Question 6.
Your mailbox is 11 paper clips long. Is your mailbox more than or less than 11 color tiles long?

Circle:       more than 11         less than 11
Tell how you know:
Answer:
My mailbox will have more than 11 color tiles long.

Explanation:
My mailbox will have more than 11 color tiles. As we can see from the above problems that while measuring with the paper clip we need fewer paper clips than the color tiles. And the length is doubled while measuring with color tile than the paper clip. So there will be more color tiles than the paper clips.

Measure More Lengths Practice 10.4

Measure

Question 1.

about __________ color tiles about __________ paper clips
Answer:
The length of the rocket is about six colored tiles and the length of the pencil which is measured using a paper clip is three paper clips.

Explanation:
The rocket is measured by using the colored tile. As each tile is equal to one unit. So the length of the rocket is six colored tiles, which means six units. We must take each tile without gaps or any overlaps between them. And to measure the rocket with the paper clip, we will place the paper clip without gaps or overlaps between them. And each paper clip is equal to about one unit. So the length of the rocket using the paper clip is three paper clips, which means three units. We have noticed that if we take a colored tile, then the length of the rocket will be six color tiles. And if we take a paper clip then the length of the pencil will be three paper clips. So the length is doubled while measuring with color tile than the paper clip.

Question 2.
YOU BE THE TEACHER
Your friend says the marker is more color tiles long than paper clips. Is your friend correct? Explain.

Answer:
Yes, the marker has more color tiles long than paper clips. So my friend is correct.

Explanation:
Yes, the marker has more color tiles long than paper clips. As we can see from the above problems that while measuring with the paper clip we need fewer paper clips than the color tiles. And the length is doubled while measuring with color tile than the paper clip. So there will be more color tiles than the paper clips for the marker. So my friend is correct.

Question 3.
Modeling Real Life
Your folder is 15 color tiles long. Is your folder more than or less than 15 paper clips long?

Circle:       more than 15         less than 15
Tell how you know:

Answer:
My folder will be less than 15 paper clips.

Explanation:
My folder will be less than 15 paper clips. As we can see from the above problems that while measuring with the paper clip we need fewer paper clips than the color tiles. And the length is doubled while measuring with color tile than the paper clip. So there will be fewer than 15 paper clips than the color tiles for my folder.

Review & Refresh

Question 4.
8 tigers swim.
5 tigers leave.
How many tigers are left?
___________ – ___________ = ___________ tigers
Answer:
the number of tigers left is
8 – 5= 3 tigers.

Explanation:
The number of tigers swim is 8 and the number of tigers which leave is 5 tigers. So the number of tigers left is
8 – 5= 3 tigers.

Question 5.
You have 6 pencils.
You lose 2 pencils.
How many pencils are left?
___________ – ___________ = ___________ pencils
Answer:
the remaining pencils are
6 – 2= 4 pencils.

Explanation:
The number of pencils I have is 6 pencils, and the number of pencils lost is 2 pencils. So the remaining pencils are
6 – 2= 4 pencils.

Lesson 10.5 Solve Compare Problems Involving Length

Explore and Grow

Draw a line that is 2 color tiles longer than the pencil.

Draw a line that is 2 color tiles shorter than the pencil.

Answer:

Show and Grow

Question 1.
Your lunch box is 6 paper clips long. Your friend’s is 3 paper clips long. How many paper clips longer is your lunch box?


__________ paper clips
Answer:
My lunch box is three paper clips longer than my friend’s and my friend’s lunch box is three times shorter than my friend’s lunch box.

Explanation:

Given that my lunch box is six paper clips long and my friend’s lunch box is three paper clips long. So there will be three paper clips longer than my friend’s. And my friend’s lunch box is three paper clips shorter than my lunch box.

Apply and Grow: Practice

Question 2.
Your scarf is 10 paper clips long. Your friend’s is 7 paper clips long. How many paper clips longer is your scarf?


Answer:
Myscarf is three paper clips longer than my friend’s and my friend’s scarf is three times shorter than my friend’s scarf.

Eplanation:

Given that my scarf is ten paper clips long and my friend’s scarf is seven paper clips long. So there will be three paper clips longer than my friend’s. And my friend’s scarf is three paper clips shorter than my scarf.

Question 3.
Your marker is 6 color tiles long. Your friend’s is 7 color tiles long. How many tiles shorter is your marker?

Answer:
My marker is one tile shorter than my friend’s marker.

Explanation:

Given that my marker is 6 color tiles long and my friend’s marker is 7 color tiles. So my marker is 7 – 6 which is one color tile shorter than my friend’s. And my friend’s marker is one color tile longer than mine.

Question 4.
MP Reasoning
Your pencil is 4 color tiles long. Your
friend’s is 2 color tiles long. Complete the sentences.

Your pencil is ________ color tiles ________ than your friend’s.
Your friend’s pencil is ________ color tiles ________ than yours.
Answer:
Your pencil is two color tiles longer than your friend’s.
Your friend’s pencil is two color tiles shorter than yours.

Explanation:
Given that my pencil is 4 color tiles long and my friend’s pencil is 2 color tiles. So my pencil is 4 – 2 which is two color tiles longer than my friend’s. And my friend’s pencil is two color tiles shorter than mine.

Think and Grow: Modeling Real Life

Your friend’s paper chain is 6 paper clips shorter than yours. Your chain is 12 paper clips long. How long is your friend’s?

Model:

Equation:

____________ paper clips long
Answer:
My paper chain is six color tiles longer than my friend’s.

Explanation:

Given that my paper chain is 12 color tiles long and my friend’s paper chain is 6 color tiles. So my paper chain is 12 – 6 which is six color tile shorter than my friend’s. And my friend’s paper chain is six color tile longer than mine.

Show and Grow

Question 5.
Your paper airplane is 9 color tiles shorter than your friend’s. Your friend’s paper airplane is 16 color tiles long. How long is yours?

Model:

Answer:
Equation:

____________ color titles long
Answer:
My airplane is seven color tiles longer than my friend.

Explanation:

Given that my airplane is 9 color tiles long and my friend’s airplane is 16 color tiles. So my marker is 16 – 9 which is seven color tile shorter than my friend’s. And my friend’s airplane is seven color tile longer than mine.

Solve Compare Problems Involving Length Practice 10.5

Question 1.
Your backpack is 15 paper clips long. Your friend’s is 12 paper clips long. How many paper clips longer is your backpack?


Answer:
Three paper clips longer than my friend’s

Explanation:

Given that my backpack is 15 paper clips long and my friend’s backpack is 12 paper clips long. So there will be three paper clips longer than my friend’s.

Question 2.
MP Reasoning
Your baseball miff is 8 paper clips long. Your friend’s is 7 paper clips long. Complete the sentences.

Your friend’s baseball mitt is ________ paper clip _________ than yours.
Your baseball mitt is _________ paper clip _________ than your friends.
Answer:
Your friend’s baseball mitt is 1 paper clip shorter than yours.
Your baseball mitt is 1 paper clip longer than your friends.

Explanation:
Given that my baseball miff is 8 paper clips long and my friend’s baseball miff is 7 paper clips. So my baseball miff is 8 – 7 which is 1 paper clip longer than my friend’s. And my friend’s baseball miff is 1 paper clip shorter than mine.

Question 3.
Modeling Real Life
Your desk is 7 paper clips longer than your friend’s. Your friend’s desk is 14 paper clips long. How long is yours?

___________ paper clips long
Answer:
7 paper clips long.

Explanation:

As my desk is 7 paper clips longer than my friend’s and my friend’s desk is 14 paper clips long. So my desk will be 14 – 7= 7 paper clips long.

Review & Refresh

Use the picture to complete the number bond.

Question 4.

Answer:
By number bond, the total number of flowers is 5 + 3= 8 flowers.

Explanation:

This number bond explains to us how the numbers are joined together and how they break down into a certain number of parts. In the above image, we can see that there are five yellow flowers and three red flowers. By number bond, we will add all the flowers. So the total number of flowers is 5 + 3= 8 flowers. Now they break down by five yellow flowers in one circle and three red flowers in another circle.

Question 5.

Answer:
By number bond, the total number of cans is 3 + 3= 8 cans.

Explanation:

This number bond explains to us how the numbers are joined together and how they break down into a certain number of parts. In the above image, we can see that there are three green cans and three red cans. By number bond, we will add all the cans. So the total number of cans is 3 + 3= 6 flowers. Now they break down by three green cans in one circle and three red cans in another circle.

Measure and Compare Lengths Performance Task

Question 1.
Use a piece of string to compare the routes from your house to the library, the post office, and the school.Order the routes from shortest to longest.
____________, ____________, ____________
Answer:
The order from shortest to longest is the post office, the library, and the school.

Explanation:
The route from my house to the library is three meters and the route from my house to the post office is two meters and the route from my house to the school is four meters. So the order from shortest to longest is the post office, the library, and the school.

Question 2.
Use a piece of string to measure the different routes from your house to your friend’s house. Color the route you would use to ride your bike to your friend’s house.
Answer:

Question 3.
a. The bakery is farther from your house than the pool. The park is closer to your house than the pool. Which place is closest to your house?
Park             Bakery              Pool
Answer:

b. Label the park, bakery, and pool on the map.
Answer:

Measure and Compare Lengths Chapter Practice

Order Objects by Length Homework & Practice 10.1

Question 1.
Order from longest to shortest.

____________, _____________, ______________
Answer:
The order from longest to shortest is
Shark, Lobster, Fish.

Explanation:
In the above image, the longest is the shark, the largest is the lobster and the shortest is the fish.

Question 2.
MP Problem Solving
A green snake is shorter than a black snake. A brown snake is shorter than a black snake. Which snake is the longest?
green        block         brown
Answer:
The black snake is the longest.

Explanation:
Given that a green snake is shorter than a black snake and a brown snake is shorter than a black snake. So the longest snake is a black snake.

Compare Lengths Indirectly Homework & Practice 10.2

Question 3.
Circle the longer object.

Answer:
We will circle the second image.

Explanation:

In the above image, the longest object is the second image. So we will circle the second image.

Question 4.
Circle the longer Object.

Answer:
We will circle the second image.

Explanation:

In the above image, the longest object is the second image. So we will circle the second image.

Question 5.
Draw a line through the shorter object.

Answer:
The first image is the shorter image.

Explanation:

In the above image, the shorter object is the first image. So we will circle the first image.

Measure Lengths Homework & Practice 10.3

Measure

Question 6.

about ___________ color tiles
Answer:
The length of the object is about four colored tiles.

Explanation:
The object is measured by using the colored tile. As each tile is equal to one unit. So the length of the object is about four colored tiles, which means four units. We must take each tile without gaps or any overlaps between them.

Question 7.

about ___________ color tiles
Answer:
The length of the object is about five colored tiles.

Explanation:
The object is measured by using the colored tile. As each tile is equal to one unit. So the length of the object is about five colored tiles, which means five units. We must take each tile without gaps or any overlaps between them.

Measure More Lengths Homework & Practice 10.4

Measure

Question 8.

about ___________ color tiles
about ___________ paper clips
Answer:
The length of the object is about four colored tiles and the length of the object which is measured using a paper clip is two paper clips.

Explanation:
The object is measured by using the colored tile. As each tile is equal to one unit. So the length of the object is four colored tiles, which means four units. We must take each tile without gaps or any overlaps between them. And to measure the object with the paper clip, we will place the paper clip without gaps or overlaps between them. And each paper clip is equal to about one unit. So the length of the object using the paper clip is two paper clips, which means two units. We have noticed that if we take a colored tile, then the length of the object will be four color tiles. And if we take a paper clip then the length of the pencil will be two paper clips. So the length is doubled while measuring with color tile than the paper clip.

Question 9.
Modeling Real Life
Your hockey stick is 18 paper clips long. Is your hockey stick more than or less than 18 color tiles long?

Circle: more than 18 less than 18
Tell how you know:
Answer:
My hockey stick has more than 18 color tiles.

Explanation:
My hockey stick has more than 18 color tiles. As we can see from the above problems that while measuring with the paper clip we need fewer paper clips than the color tiles. And the length is doubled while measuring with color tile than the paper clip. So there will be more than 18 color tiles than the paper clips for the hockey stick.

Solve Compare Problems Involving Length Homework & Practice 10.5

Question 10.
Your water bottle is 5 paper clips long. Your Friend’s is 4 paper clips long. How many paper clips longer is your water bottle?

Answer:
One paper clip long.

Explanation:

Given that my water bottle is 5 paper clips long and my friend’s water bottle is 4 paper clips long. So there will be one paper clip longer than my friend’s.

Question 11.
Your bookshelf is 19 color tiles long. Your friend’s is 15 color tiles long. How many tiles longer is your bookshelf?


Answer:
My bookshelf is four tiles longer.

Explanation:

Given that my bookshelf is 19 paper clips long and my friend’s bookshelf is 15 paper clips long. So there will be four paper clips longer than my friend’s.

Question 12.
MP Reasoning
Your pencil is 6 color tiles long. Your friend’s is 3 color tiles long. Complete the sentences.

Your pencil is ____________ color tiles ____________ than your friend’s.
Your friend’s pencil is ____________ color tiles ____________ than yours.
Answer:
Your pencil is three color tiles longer than your friend’s.
Your friend’s pencil is three color tiles shorter than yours.

Explanation:
As my pencil is six color tiles long, and my friend’s pencil is three color tiles long. So my pencil will be three color tiles longer than my friend’s pencil. And my friend’s pencil is three color tiles shorter than my pencil.

Conclusion:

Hope the information regarding Big Ideas Math Answers Grade 1 Chapter 10 Measure and Compare Lengths is beneficial for you. If you have any doubts please post the comments in the below-mentioned comment box. Share this pdf link with your friends and help them to overcome the difficulties in maths.

Big Ideas Math Book 3rd Grade 15th Chapter Find Perimeter and Area Answer Key

Students have to prepare all the topics of Big Ideas Math Answers 3rd Grade 15th Chapter Find Perimeter and Area to score good marks in the exam. The different topics included in BIM Book Grade 3 Chapter 15 Find Perimeter and Area Answers are Understand Perimeter, Find Perimeters of Polygons, Find Unknown Side Lengths, Same Perimeter, Different Areas, and Same Area, Different Perimeters. After preparing these topics, check your skills at the performance task provided at the end.

The success criteria of Find Perimeter and Area is to understand the area and perimeter of polygons. The topic wise quick links to access Big Ideas Math Book Grade 3 Chapter 15 Find Perimeter and Area Answer Key is provided below. So, just tap on them and get the solutions easily.

Lesson 1 Understand Perimeter

• Lesson 15.1 Understand Perimeter
• Understand Perimeter Homework & Practice 15.1

Lesson 2 Find Perimeters of Polygons

• Find Perimeters of Polygons 15.2
• Find Perimeters of Polygons Homework & Practice 15.2

Lesson 3 Find Unknown Side Lengths

• Lesson 15.3 Find Unknown Side Lengths
• Find Unknown Side Lengths Homework & Practice 15.3

Lesson 4 Same Perimeter, Different Areas

• Lesson 15.4 Same Perimeter, Different Areas
• Same Perimeter, Different Areas Homework & Practice 15.4

Lesson 5 Same Area, Different Perimeters

• Lesson 15.5 Same Area, Different Perimeters
• Same Area, Different Perimeters Homework & Practice 15.5

Performance Task

• Find Perimeter and Area Performance Task
• Find Perimeter and Area Activity
• Find Perimeter and Area Chapter Practice
• Find Perimeter and Area Cumulative practice 1 – 15
• Find Perimeter and Area Cumulative Steam Performance Task 1 – 15

Lesson 15.1 Understand Perimeter

Explore and Grow

Question 1.
Model a rectangle on your geoboard. Draw the rectangle and label its side lengths. What is the distance around the rectangle?

Answer:
The units around the rectangle are 15 units.

Explanation:

In the above figure, the length of the rectangle is 4 units, and
the breadth of the rectangle is 3 units.
The units around the rectangle are 15 units.

Structure
Change the side lengths of the rectangle on your geoboard. What do you notice about the distance around your rectangle compared to the distance around the rectangle above? Explain

Answer:
The distance around the rectangle is 15 units

Explanation:

In the above figure, we can see the length of the rectangle is 3 units, and
the breadth of the rectangle is 4 units.
There is no change in the distance around the rectangle which is 15 units.

Think and Grow: Understand Perimeter
Perimeter is the distance around a figure. You can measure perimeter using standard units, such as inches, feet, centimeters, and meters.
Example
Find the perimeter of the rectangle.
Choose a unit to begin counting. Count each unit around the rectangle.

Answer:
The perimeter of the rectangle is 20 in.

Explanation:
To find the perimeter of the triangle, we need to know the length and the breadth of the rectangle.
Here, the length of the rectangle is 7 in,
and the breadth is 3 in,
So the perimeter of the rectangle is 2(Length + Breadth)
= 2(7+3)
= 2(10)
= 20 in.

Show and Grow

Question 1.
Find the perimeter of the figure.

There are ___ units around the figure.
Answer:
There are 22 units around the figure.

Explanation:
To find the perimeter of the rectangle, we need to know the length and the breadth of the rectangle.
Here, the length of the rectangle is 6 m,
and the breadth is 5 m,
So the perimeter of the rectangle is 2(Length + Breadth)
= 2(6+5)
= 2(11)
= 22 m.

Question 2.

There are ___ units around the figure.
So, the perimeter is ____ feet.
Answer:
The perimeter of the figure is 24 ft.

Explanation:
To find the perimeter of the figure, we will add all the sides of the figure
So the sides of the figure are 5,3,2,4,3,7.
and the perimeter of the figure is 5+3+2+4+3+7
= 24 ft.

Question 3.
Draw the figure that has a perimeter of 16 centimeters.

Answer:
The sides of the figure are 4 cm.

Explanation:
As the perimeter of the figure is 16 cm,
So let the sides of the figure be 4 cm,
by that, we can get the perimeter as 16 cm, i.e
4+4+4+4= 16 cm.

Apply and Grow: Practice

Question 4.
Find the perimeter of the figure.

There are ___ units around the figure.
So, the perimeter is ___
Answer:
There are 20 units around the figure.
So, the perimeter is 20 cm.

Explanation:
To find the perimeter of the figure, we will add all the sides of the figure
So the sides of the figure are 3,3,2,2,5,5.
and the perimeter of the figure is 3+3+2+2+5+5
= 20 units,
So the perimeter is 20 cm.

Question 5.

There are ___ units around the figure.
So, the perimeter is ___
Answer:
There are 26 units around the figure.
So, the perimeter is 26 ft.

Explanation:
To find the perimeter of the figure, we will add all the sides of the figure
So the sides of the figure is 5,5,2,3,1,3,2,5
and the perimeter of the figure is 5+5+2+3+1+3+2+5
= 26 units,
So the perimeter is 26 ft.

Question 6.

Perimeter = ___
Answer:
The perimeter of the figure is 22 in.

Explanation:
To find the perimeter of the figure, we will add all the sides of the figure
So the sides of the figure is 4,3,2,3,3,3,1,3
and the perimeter of the figure is 4+3+2+3+3+3+1+3
= 22 units,
So the perimeter is 22 in.

Question 7.

Perimeter = ___
Answer:
The perimeter is 24 m.

Explanation:
To find the perimeter of the figure, we will add all the sides of the figure
So the sides of the figure is 5,7,2,2,1,2,1,1,1,2
and the perimeter of the figure is 5+7+2+2+1+2+1+1+1+2
= 24 units,
So the perimeter is 24 m.

Question 8.
Draw a figure that has a perimeter likely measurement for the of 14 centimeters.

Answer:
The length of the rectangle is 2 cm, and
the breadth is 5 cm.

Explanation:

In the above figure, we can see the length of the rectangle is 2 cm
and the breadth of the rectangle is 5 cm
perimeter = 2(length + breadth)
= 2(5+2)
= 2(7)
= 14 cm.
so the perimeter of the rectangle is 14 cm.

Question 9.
Precision Which is the most likely measurement for the perimeter of a photo?
20 inches
100 meters
5 centimeters
2 inches
Answer:
2 inches.

Explanation:
The most likely measurement for the perimeter of a photo is 2 inches.

Question 10.
You be the teacher Your friend counts the units around the figure and says the perimeter is 12 units. Is your friend correct? Explain.

Answer:
No, he is not correct.

Explanation:
No, he is not correct. As there are 2+2+1+1+4+1+1+2= 14 units, but not 12 units. So he is not correct.

Think and Grow: Modeling Real Life
Use a centimeter ruler to find the perimeter of the bookmark.

The perimeter is ___
Answer:
The perimeter of the bookmark is 26 cm.

Explanation:
On measuring, the length of the sides of the bookmark are 4 cm, 9 cm, 2 cm, 2 cm, 9 cm
to find the perimeter of the bookmark, we will add all the length of the sides
so the perimeter of the bookmark is
p = 4 cm+ 9 cm+ 2 cm+ 2 cm+ 9 cm
= 26 cm.
The perimeter of the bookmark is 26 cm.

Show and Grow

Question 11.
Use an inch ruler to find the perimeter of the decal.

Answer:
The perimeter of the decal is 17 in.

Explanation:
On measuring, the length of the sides of the decal is 5in, 2in, 5in, 5in
to find the perimeter of the decal, we will add all the length of the sides
so the perimeter of the decal is
p= 5in+2in+5in+5in
= 17 in
The perimeter of the decal is 17 in.

Question 12.
How much greater is the perimeter of your friend’s desk than the perimeter of your desk?

Answer:
Friend’s perimeter is 22-16= 6 times greater.

Explanation:
As we can see figure 1 is a square and the sides of the square are 4 ft.
So the perimeter of the square is a+a+a+a,
= 4+4+4+4
= 16 ft.
And now let’s find the perimeter of the friend’s figure,
So the sides of the figure is 2,6,5,2,3,4
and the perimeter is 2+6+5+2+3+4= 22 ft.
By this, we can see that the friend’s figure has a greater perimeter,
and friend’s perimeter is 22-16= 6 times greater.

Question 13.
DIG DEEPER!
Explain how you might use a centimeter ruler and string to estimate the perimeter of the photo of the window.

Answer:
Here, we will use the ruler to find the length of the bottom part of the window and the sides which are straight we can find the length using the ruler. And the curve sides we will measure using the string.

Understand Perimeter Homework & Practice 15.1

Question 1.

There are ___ units around the figure.
So, the perimeter is ___ inches.
Answer:
There are 20 units around the figure,
So the perimeter is 20 in.

Explanation:
To find the perimeter of the figure, we will add all the sides of the figure,
As we can see the above image is a square,
and the perimeter of the square is s+s+s+s
= 5+5+5+5
= 20 in.
As there are 20 units around the figure,
So the perimeter is 20 in.

Question 2.

There are ___ units around the figure.
So, the perimeter is ___ feet.

Answer:
There are 32 units around the figure,
So the perimeter is 32 ft.

Explanation:
To find the perimeter of the figure, we will add all the sides of the figure,
The sides of the above figure is 2,3,4,3,2,5,8,5
and the perimeter of the figure is 2+3+4+3+2+5+8+5
= 32 units.
As there are 32 units around the figure,
So the perimeter is 32 ft.

Question 3.

Perimeter = ___
Answer:
There are 26 units around the figure,
So the perimeter is 26 cm.

Explanation:
To find the perimeter of the figure, we will add all the sides of the figure,
The sides of the above figure is 2,2,2,3,1,3,5,8
and the perimeter of the figure is 2+2+2+3+1+3+5+8
= 26 units.
As there are 26 units around the figure,
So the perimeter is 26 cm.

Question 4.

Perimeter = ___
Answer:
There are 38 units around the figure,
So the perimeter is 38 m.

Explanation:
To find the perimeter of the figure, we will add all the sides of the figure,
The sides of the above figure is 2,8,2,3,1,3,2,8,2,3,1,3
and the perimeter of the figure is 2+8+2+3+1+3+2+8+2+3+1+3
= 38 units.
As there are 38 units around the figure,
So the perimeter is 38 m.

Question 5.
Draw a figure that has a perimeter of 18 inches.

Answer:
The length of the rectangle is 5 in
and the breadth of the rectangle is 4 in.

Explanation:

Let the length of the rectangle be 5 in
and the breadth of the rectangle be 4 in
so the perimeter of the rectangle is
p = 2(length+breadth)
= 2(5+4)
= 2(9)
= 18 in.

Question 6.
Reasoning
Which color represents the perimeter of the rectangle? What does the other color represent?

Answer:
Blue color represents the perimeter of the rectangle and
the other color yellow represents the area of the figure.

Explanation:
In the above figure, the blue color represents the perimeter of the rectangle.
Because the perimeter represents the distance around the edge of the shape.
And the other color yellow represents the area of the figure,
as area represents the amount of space inside a shape.

Question 7.
Modeling Real Life
Use a centimeter ruler tofind the perimeter of the library card.

Answer:
The perimeter of the library card is 24 cm.

Explanation:
On measuring, the length of the sides of the library card is 5cm, 7cm, 5cm, 7cm
to find the perimeter of the library card, we will add all the length of the sides
so the perimeter of the library card is
p= 5cm+7cm+5cm+7cm
= 24 cm
The perimeter of the library card is 24 cm.

Question 8.
Modeling Real Life
How much greater is the perimeter of your piece of fabric than the perimeter of your friend’ spiece of fabric?

Answer:
My fabric is 4 inches greater than my friend’s fabric.

Explanation:
To find the perimeter of the figure 1, we will add all the sides of the figure,
The sides of the above figure is 2,1,1,2,2,2,8,2,2,2,1,1
and the perimeter of the figure is 2+1+1+2+2+2+8+2+2+2+1+1
= 26 in.
So the perimeter of the figure is 26 in.
And now let’s find the perimeter of the friend’s figure,
So the sides of the figure is 1,2,2,4,1,5,4,3
and the perimeter is 1+2+2+4+1+5+4+3
= 22 in.
By this, we can see that my fabric has the highest perimeter than the friend’s fabric
which is 26-22= 4 in greater.

Review & Refresh

Write two equivalent fractions for the whole number.

Question 9.

Answer:
1= 4/4 = 6/6

Explanation:
Here, the equivalent fraction for the whole number means if the numerator was divided by the denominator without any reminder then the fraction is equivalent to a whole number. So 1= 4/4 = 6/6.

Question 10.

Answer:
4= 4/1 = 8/2.

Explanation:
Here, the equivalent fraction for the whole number means if the numerator was divided by the denominator without any reminder then the fraction is equivalent to a whole number. So 4= 4/1 = 8/2.

Question 11.

Answer:
6= 24/4 = 36/6.

Explanation:
Here, the equivalent fraction for the whole number means if the numerator was divided by the denominator without any reminder then the fraction is equivalent to a whole number. So 6= 24/4 = 36/6.

Find Perimeters of Polygons 15.2

Explore and Grow

Model a rectangle on your geoboard. Draw the rectangle and label its side lengths. Then find the perimeter in more than one way.

Answer:
The perimeter of the rectangle is 14 units.

Explanation:

In  the above figure, we can see the length of the rectangle is 4 units
and the breadth of the rectangle is 3 units,
so the perimeter of the rectangle is
p =2(length + breadth)
= 2(4+3)
= 2(7)
= 14 units.
The perimeter of the rectangle is 14 units.
Critique the Reasoning of Others
Compare your methods of finding the perimeter to your partner’s methods. Explain how they are alike or different.
Answer:
There are two methods to find the perimeter explained below.

Explanation:
There are two ways to find the perimeter.
The first method is
Perimeter = 2(length + breadth)
here, we will add length and breadth, and then we will multiply the result by 2.
and the second method is
Perimeter = a+b+c+d
here, we will add all the sides of the figure.

Think and Grow: Find Perimeter

Example
Find the perimeter of the trapezoid.

You can find the perimeter of a figure by adding all of the side lengths.

Write an equation. The letter P represents the unknown perimeter. Add the side lengths.
So, the perimeter is ___.
Answer:
So, the perimeter is 36 in.

Explanation:
To find the perimeter of the trapezoid, we will add all the sides of the trapezoid,
so the sides of the trapezoid is 6 in,12 in,8 in,10 in
The perimeter is 6+12+8+10
= 36 in.

Example
Find the perimeter of the rectangle

Because a rectangle has two pairs of equal sides, you can also use multiplication to solve.

One Way:

Another Way:

So, the perimeter is ___.

Answer:
Perimeter = 7+9+7+9
= 32 cm.
Perimeter= 2×9 + 2× 7
= 32 cm.
So, the perimeter is 32 cm.

Explanation:
To find the perimeter of a rectangle, we have two ways,
One way is to add all the sides of the rectangle, which is
7+9+7+9= 32 cm.
And the other way is, as the two sides of the rectangle are equal, we wil use formula
Perimeter = 2( Length + Breadth)
= 2× Length + 2 × Breadth
= 2×9 + 2× 7
= 18+ 14
= 32 cm.

Show and Grow

Find the perimeter of the polygon.
Answer:

Explanation:
To find the perimeter of the polygon, we will add the length of the all sides.

Question 1.

The perimeter is ___.
Answer:
The perimeter is 12 m.

Explanation:
To find the perimeter of the polygon, we will add the length of all sides of the polygon.
So the length of the sides is 5 m, 3 m, 3 m, 2 m
The perimeter is 5+3+3+2
= 12 m.

Question 2.

The perimeter is ___.

Answer:
The perimeter is 24 ft.

Explanation:
As we can see in the above figure which has all sides are equal,
so the perimeter of the square is 4s
= 4×6
= 24 ft.

Apply and Grow: Practice

Find the perimeter of the polygon.

Question 3.

Perimeter = ___
Answer:
The perimeter of the polygon is 28 m.

Explanation:
To find the perimeter of the polygon, we will add the length of the all sides of the polygon.
So the length of the sides is 4 m, 9 m, 10 m, 5 m
The perimeter is 4+9+10+5
= 28 m.

Question 4.

Perimeter = ___
Answer:
The perimeter of the figure is 40 in.

Explanation:
To find the perimeter of the figure, we will add the length of all sides of the figure.
So the length of the sides is 12 in, 4 in, 8 in, 7 in,9 in.
The perimeter is 12+4+8+7+9
= 40 in.

Question 5.
Rectangle

Perimeter = ___
Answer:
The perimeter of the rectangle is 36 cm.

Explanation:
To find the perimeter of the rectangle, we need to know the length and the breadth of the rectangle.
Here, the length of the rectangle is 9 cm,
and the breadth is 10 cm,
So the perimeter of the rectangle is 2(Length + Breadth)
= 2(10+9)
= 2(19)
= 38 cm.

Question 6.
Rhombus

Perimeter = ___
Answer:
The perimeter of the Rhombus is 4 in.

Explanation:
As all sides of the Rhombus are equal, so the formula of the Rhombus is 4a
and the side of the Rhombus is 1 in,
so the perimeter is 4a
= 4×1
= 4 in.

Question 7.
Parallelogram

Perimeter = ___
Answer:
The perimeter of the parallelogram is 22 cm.

Explanation:
As the opposite sides of the parallelogram are equal,
so the perimeter of the parallelogram is 2(a+b)
the length of the parallelogram is 3 cm,
and the breadth of the parallelogram is 8 cm
So the perimeter of the parallelogram is
= 2(3+8)
= 2(11)
= 22 cm.

Question 8.
Square

Perimeter = ___
Answer:
The perimeter of the square is 16 ft.

Explanation:
As we can see in the above figure which has all sides are equal,
so the perimeter of the square is 4s
= 4×4
= 16 ft.

Question 9.
You build a pentagon out of wire for a social studies project. Each side is 8 centimeters long. What is the perimeter of the pentagon?
Answer:
The perimeter of the pentagon is 40 cm.

Explanation:
The length of the pentagon is 8 cm,
so the perimeter of the pentagon is 5a,
which is 5×8
= 40 cm

Question 10.
Number Sense
The top length of the trapezoid is 4 feet. The bottom length is double the top. The left and right lengths are each 2 feet less than the bottom. Label the side lengths and find the perimeter of the trapezoid.

Answer:
The sides of the trapezoid are 4ft, 6ft, 8ft, 6ft, and the perimeter is 24 ft.

Explanation:
As the top length of the trapezoid is 4 feet and the bottom length is double the top,
which is 4×2= 8 feet. And the left and right lengths are each 2 feet less than the bottom,
which means 8 – 2 = 6 ft each.

So to find the perimeter of the trapezoid is we will add the length of all sides.
The perimeter of the trapezoid is 4+6+8+6= 24 ft.

Question 11.
Writing
Explain how finding the perimeter of a rectangle is different from finding its area.
Answer:
Perimeter= 2(length + breadth)
Area = length × breadth.

Explanation:
To find the perimeter of the rectangle,
we will add the length and breadth and will multiply the result with 2
Perimeter= 2(length + breadth)
and to find the area of the rectangle,
we will multiply the length and breadth of the rectangle.
Area = length × breadth.

Question 12.
Dig Deeper!
A rectangle has a perimeter of 12 feet. What could its side lengths be ?
Answer:
The length of the rectangle is 4 feet
and the breadth of the rectangle is 2 feet

Explanation:
Given the perimeter is 12 feet,
so the let the length be 4 feet
and the breadth be 2 feet
Let’s check on the length and breadth is correct are not
perimeter = 2( length + breadth)
= 2( 4+2)
= 2(6)
= 12 feet.

Think and Grow: Modeling Real Life

The rectangular sign is 34 feet longer than it is wide. What is the perimeter of the sign?

Understand the problem:
Make a plan:
Solve:
The perimeter is ___.
Answer:
The perimeter of the rectangular sign is 124 ft.

Explanation:
Given the rectangular sign is 34 feet longer than it’s wide
and the wide is 14 ft,
so the length is 34ft + 14ft= 48 ft.
the perimeter of the rectangular sign is
P= 2(48ft + 14ft)
= 2(62ft)
= 124 ft.
The perimeter of the rectangular sign is 124 ft.

Show and Grow

Question 13.
A city has a rectangular sidewalk in a park. The sidewalk is 4 feet wide and is 96 feet longer than it is wide. What is the perimeter of the sidewalk?
Answer:
The perimeter of the sidewalk is 208 feet.

Explanation:
As given the rectangular sidewalk’s wide is 4 feet and the length is 96 feet longer than it’s wide,
which means 96+4= 100 feet is the length of the rectangular sidewalk,
so the perimeter of the rectangular sidewalk is 2( length + breadth)
= 2( 4+100)
= 2(104)
= 208 feet.

Question 14.
A team jogs around a rectangular field three times. The field is 80 yards long and 60 yards wide. How many yards does the team jog?
Answer:
The number of yards does the team jog is 3×280= 840 yards.

Explanation:
The length of the rectangular field is 80 yards,
The breadth of the rectangular field is 60 yards,
So, the perimeter of the rectangular field is 2(length + breadth)
= 2(80+60)
= 2(140)
= 280 yards.
As the team jogs around a rectangular field three times,
so the number of yards does the team jog is 3×280= 840 yards.

Question 15.
Each side of the tiles is 8 centimeters long. What is the sum of the perimeters?

You put the tiles together as shown. Is the perimeter of this new shape the same as the sum of the perimeters above? Explain.

Answer:
The sum of the perimeters is 96 cm.
Yes, there will be a change in the perimeter of the new figure. The perimeter is 80 cm.

Explanation:
In the above figure, we can see a hexagon that has six sides.
So the formula for the perimeter of a hexagon is 6a,
the perimeter of the tiles is 6×8
= 48 cm.
So the sum of the tiles 48+48= 96 cm.
Yes, there will be a change in the perimeter of the new figure. As there are ten sides in the new figure, so the perimeter of the new figure is 10×8= 80 cm.

Find Perimeters of Polygons Homework & Practice 15.2

Find the perimeter of the polygon

Question 1.

Perimeter = ___
Answer:
The perimeter of the polygon is 29 in.

Explanation:
The perimeter of the polygon is the sum of the length of its sides,
So the sides of the polygon are 9 in, 4 in, 6 in, 10 in,
and the perimeter of the polygon is 9+4+6+10= 29 in.

Question 2.

Perimeter = ___
Answer:
The perimeter of the figure is 38 cm.

Explanation:
The perimeter of the figure is the sum of the length of its sides,
So the sides of the figure are 6 cm, 5 cm, 7 cm, 8 cm, 7 cm, 5 cm,
and the perimeter of the figure is 6+5+7+8+7+5= 38 cm.

Question 3.
Square

Perimeter = ___
Answer:
The perimeter of the square is 8 ft.

Explanation:
The length of the square is 2 ft,
and the perimeter of the square is 4a
= 4×2
= 8 ft.
so the perimeter of the square is 8 ft.

Question 4.
Parallelogram

Perimeter = ___
Answer:
The perimeter of the parallelogram is 8m.

Explanation:
The length of the parallelogram is 1 m,
and the breadth of the parallelogram is 3 m,
so the perimeter of the parallelogram is 2(length + breadth)
= 2(1+3)
= 2(4)
= 8 m.

Question 5.
Rhombus

Perimeter = ___
Answer:
The perimeter of the rhombus is 40 ft.

Explanation:
The length of the side of the rhombus is 10 ft,
and the perimeter of the rhombus is 4a
= 4×10
= 40 ft.

Question 6.

Perimeter = ___
Answer:
The perimeter of the rectangle is 22 in.

Explanation:
To find the perimeter of the rectangle, we need to know the length and the breadth of the rectangle.
Here, the length of the rectangle is 7 in,
and the breadth is 4 in,
So the perimeter of the rectangle is 2(Length + Breadth)
= 2(7+4)
= 2(11)
= 22 in.

Question 7.
Each side of a triangle is 5 centimeters long. What is the perimeter of the triangle?
Answer:
The perimeter of the triangle is 15 cm.

Explanation:
Given the length of the side of the triangle is 5 cm,
and the perimeter of the triangle is 3a
= 3×5
= 15 cm.

Question 8.
You Be The Teacher
Descartes says that a square will always have a greater perimeter than a triangle because it has more sides. Is he correct? Explain.
Answer:
Yes, Descartes is correct.

Explanation:
Yes, Descartes is correct. As the square has four sides and the perimeter of the square is 4a.
Whereas the triangle has 3 sides and the perimeter of the triangle is 3a.
So the square will always have a greater perimeter than a triangle.

Question 9.
Structure
Draw a pentagon and label its sides so that it has the same perimeter as the rectangle.

Answer:
The length of the side of the pentagon is 4m.

Explanation:
Given the length of the rectangle is 7 m,
and the breadth of the rectangle is 3 m,
So the perimeter of the rectangle is 2( length + breadth)
= 2(7+3)
= 2(10)
= 20 m.
Here we have the perimeter of the pentagon which is 20 m,
so we should find the sides of the pentagon,
As we know the perimeter of the pentagon is 5a
5a= 20
a= 20/5
= 4 m.
So the length of the side of the pentagon is 4m.

Question 10.
Modeling Real Life
An Olympic swimming pool is 82 feet longer than it is wide. What is the perimeter of the swimming pool?

Answer:
The perimeter of the swimming pool is 492 ft.

Explanation:
Given the Olympic swimming pool is 82 feet longer than it’s wide
and the wide is 82 ft,
so the length of the swimming pool is 82+82= 164 ft
the perimeter of the swimming pool is
p= 2(length+breadth)
= 2(164+82)
= 2(246)
= 492 ft.
The perimeter of the swimming pool is 492 ft.

Question 11.
Modeling Real Life
You put painter’s tape around two rectangular windows. The windows are each 52 inches long and 28 inches wide. How much painter’s tape do you need?
Answer:
The painter’s tape 160 inches.

Explanation:
Given the length of the window is 52 inches
and the width of the window is 28 inches
so the perimeter of the window is
p= 2(length + breadth)
= 2(52+28)
= 2(80)
= 160 inches.
So the painter’s tape 160 inches.

Review & Refresh

Question 12.

Answer:
737.

Explanation:
On adding 590+147 we will get 737.

Question 13.

Answer:
894.

Explanation:
On adding 636+258 we will get 894

Question 14.

Answer:
805.

Explanation:
On adding 476+329 we will get 805

Lesson 15.3 Find Unknown Side Lengths

Explore and Grow

You have a map with the three side lengths shown. The perimeter of the map is 20 feet. Describe how you can find the fourth side length of your map without measuring.

Answer:
The fourth side of the map is 4ft.

Explanation:
We can find the value of the fourth side in two methods
First method:
Given the perimeter of the map is 20 feet,
and the sides of the map is 6 ft, 4 ft, 6ft, X ft.
As we know the perimeter of the rectangle is
6+4+6+X= 20 feet
16+X= 20
X= 20- 16
= 4 ft.
Second method:
As we know that the opposite sides of the rectangle are equal, as we know that the length of the side is 4 ft so the other side will also be 4 ft.

Repeated Reasoning
How is finding the unknown side length of a square different from finding the unknown side length of a rectangle?
Answer:
Refer below for a detailed explanation.

Explanation:
To find the unknown side length of the square
if we know the perimeter of the square then
the perimeter of the square is
p= 4a
we will substitute the value of p, on solving we will get the length of the square.
and to find the unknown side length of the rectangle,
we need to know the area or perimeter of the rectangle
and the other side of the rectangle.
so the formula of the perimeter of the rectangle is
p = 2(length + breadth)
we will substitute the perimeter value and the other side value
then we can find the length of the rectangle.

Think and Grow: Find Unknown Side Lengths
Example
The perimeter of the trapezoid is 26 feet. Find the unknown side length.


Write an equation for the perimeter.
Add the known side lengths.
What number plus 16 equals 26?
The unknown side length is ___.
Answer:
K= 10,
The number 16+10 equals 26.
The unknown side length is 10.

Explanation:
Given the perimeter of the trapezoid is 26 ft,
So the perimeter of the trapezoid is
K+5+6+5= 26
K+16= 26
K= 10.
The number 16+10 equals 26.
The unknown side length is 10.

Example

The perimeter of the square is 32 centimeters. Find the length of each side of the square.


Write an equation for the perimeter 4 times what number equals 32?
So, the length of each side is ___.
Answer:
n= 8.
The length of each side is 8 cm.

Explanation:
The perimeter of the square is 32 cm
So to find the sides of the square
4a= 32
a= 32/4
= 8 cm.
So, the length of each side is 8 cm.

Show and Grow

Find the unknown side length.

Question 1.
Perimeter = 34 inches

y = ___
Answer:
y = 13 in.

Explanation:
Given the perimeter is 34 inches,
and the sides of the figure are 10 in, 7 in, 4 in, y in.
so the perimeter of the figure is
34 in = 10+7+4+y
34 = 21+ y
y = 34 – 21
y = 13.

Question 2.
Perimeter = 20 meters

j = ___
Answer:
The length of the sides of the square is 5 m.

Explanation:
As we can see the above figure is a square and the perimeter of the square 20 meters,
so the sides of the square are
perimeter = 4a
20 = 4 j
j= 5 m.

Apply and Grow: Practice

Find the unknown side length.

Question 3.
Perimeter = 19 feet

y = ___
Answer:
The perimeter of the figure is 8 ft.

Explanation:
The perimeter of the figure is 19 feet,
and the length of the sides of the figure is 8 ft, 3 ft, y ft.
so perimeter = 8 ft + 3 ft + y ft
19= 11 ft + y ft
y= 19 ft – 11 ft
= 8 ft.

Question 4.
Perimeter = 26 centimeters

d = ___
Answer:
d= 4 cm.

Explanation:
The perimeter of the figure is 26 cm,
and the length of the sides of the figure is 10 cm, 5 cm, 7 cm, d cm.
so the perimeter of the figure is
p = 10+5+7+d
26 = 22 + d
d= 26-22
= 4

Question 5.
Perimeter = 30 feet

k = ___
Answer:
k = 11 ft.

Explanation:
Given the perimeter of the figure is 30 feet,
and the length of the sides is 5ft, 12 ft, 2 ft, k ft
So the perimeter of the figure is
p = 5 + 12 + 2 + k
30 ft = 19 ft + k
k = 11 ft.

Question 6.
Perimeter = 32 inches

k = ___
Answer:
k= 4 in.

Explanation:
Given the perimeter of the figure is 32 inches,
and the lengths of all sides is 10 in, 4 in, 5 in, 4 in, 5 in, k in.
So the perimeter of the figure is
p= 10+4+5+4+5+k
32 in =  28 in + k
k= 4 in.

Question 7.
Perimeter = 8 meters

y = ___
Answer:
y = 2 m.

Explanation:
Given the perimeter of the rhombus is 8 feet,
and the length of the side is y m,
So the perimeter of the rhombus is
p = 4a
8 m = 4×y
y = 2 m.

Question 8.
Perimeter = 48 inches

d = ___
Answer:
d = 8 in.

Explanation:
Given the perimeter of the Hexagon is 48 inches,
and the length of the sides is d in
So the perimeter of the hexagon is
p = 6 a
48 in = 6 × d in
d= 8 in.

Question 9.
Number Sense
A rectangle has a perimeter of 30 centimeters. The left side is 7 centimeters long. What is the length of the top side?
Answer:
The length of the top side is 8 cm.

Explanation:
Given the perimeter of the rectangle is 30 cm,
and the length of the left side of the rectangle is 7 cm,
So let the length of the top side be X,
Perimeter of the rectangle is
P = 2 (Length + breadth)
30 = 2 ( 7 cm + X cm)
30 / 2 = 7 cm + X cm
15 = 7 cm + X cm
X = 15 cm – 7 cm
X = 8 cm.
so, the length of the top side is 8 cm.

Question 10.
Writing
A triangle has three equal sides and a perimeter of 21 meters. Explain how to use division to find the side lengths.
Answer:
The length of the side is 7 m.

Explanation:
Given the perimeter of the triangle is 21 m,
and we need to find the side of the lengths,
so the perimeter of the triangle is
p = 3a
21 m = 3×a
a = 21/3
= 7 m
So the length of the side is 7 m.

Question 11.
DIG DEEPER!
Newton draws and labels the square and rectangle below. The perimeter of the combined shape is 36 feet. Find the unknown side length.

Answer:
The unknown side of the length is 14 ft.

Explanation:
As the perimeter of the combined shape is 36 feet,
and the length of the side of the rectangle is 4 ft, and the other side be X ft
and the perimeter of the rectangle is 36 ft,
so perimeter = 2 (length + breadth)
36 ft  = 2( 4 ft + X ft)
36/2 = 4 ft + X ft
18 = 4 ft + X ft
X= 14 ft.
The unknown side of the length is 14 ft.

Think and Grow: Modeling Real Life

The perimeter of the rectangular vegetable garden is 30 meters. What are the lengths of the other three sides?

Understand the problem:
Make a plan:
Solve:
The lengths of the other three sides are ___, ___, and ___.
Answer:
The lengths of other three sides is 6 m, 9 m, 9 m.

Explanation:
The perimeter of the rectangular vegetable garden is 30 m
as it is in a rectangular shape, so the opposite sides are equal,
and the length of the side of the rectangular vegetable garden is 6m,
let the other side be X m
so the perimeter is
p = 2( length + breadth)
30 m = 2( 6 m+ X m)
30/2 = 6 + X
15 = 6 + X
X= 15 – 6
= 9 m.
So the lengths of other three sides is 6 m, 9 m, 9 m.

Show and Grow

Question 12.
The perimeter of the rectangular zoo enclosure is 34 meters. What are the lengths of the other three sides?

Answer:
The lengths of the other three sides are 12 m, 11 m, 11 m.

Explanation:
The perimeter of the rectangular zoo is 34 m
as it is in rectangular shape, so the opposite sides are equal,
and the length of the side of the rectangular zoo is 12 m,
let the other side be X m
so the perimeter is
p = 2( length + breadth)
34 m = 2( 12 m+ X m)
34/2 = 12 + X
17 = 6 + X
X= 17 – 6
= 11 m.
So the lengths of the other three sides is 12 m, 11 m, 11 m.

Question 13.
The floor of an apartment is made of two rectangles. The Perimeter is 154 feet. What are the lengths of the other three sides?

Answer:
The other length of the side of the small rectangle is 7 ft.
The other length of the side of the big rectangle is 30 ft.

Explanation:
Given the perimeter of the apartment is 154 feet,
first, we will take the big rectangle,
as the opposite sides of the rectangle are equal and the length of the big rectangle is 30 ft
so the other length is also 30 ft.
as the perimeter of the small rectangle is 38 ft
and the length of the one side of the rectangle is 12 ft
so the other length of the small rectangle is
p = 2(length+breadth)
38 = 2(12 + breadth)
38/2 = 12+ breadth
19= 12 + breadth
breadth= 19 – 12
= 7 ft.
The other length of the side is 7 ft.

Question 14.
DIG DEEPER!
You want to make a flower bed in the shape of a pentagon. Two sides of the flower bed are each 7 inches long, and two sides are each 16 inches long. The perimeter is 57 inches. Sketch the flower bed and label all of the side lengths.
Answer:
The length of the other side is 11 ft.

Explanation:

Given the perimeter of the flower bed shaped pentagon is 57 inches
and the two sides of the flower bed each is 7 inches long
and the other two sides of the flower bed each is 16 inches long
the other side of the flower bed be X
the perimeter of the flower bed is
p = 7+7+16+16+X
57= 46+X
X= 11 in.
The length of the other side is 11 ft.

Find Unknown Side Lengths Homework & Practice 15.3

Find the unknown side length.

Question 1.
Perimeter = 24 feet

d = ___
Answer:
The length of the unknown side is 8 ft.

Explanation:
The perimeter of the triangle is 24 ft,
and the lengths of the sides is 10 ft, 6 ft, d ft
so the perimeter of the triangle is
p = 10+6+d
24 = 10+6+d
24 = 16+d
d = 24 – 16
=  8 ft.
The length of the unknown side is 8 ft.

Question 2.
Perimeter = 46 inches

k = ___
Answer:
The length of the unknown side is 15 in.

Explanation:
The perimeter of the figure is 46 inches,
and the lengths of the sides is 13 in, 5 in, 13 in, k in
so the perimeter of the figure is
p = 13+5+13+k
46 = 31+k
46-31 = k
k = 15
=  15 in.
The length of the unknown side is 15 in.

Question 3.
Perimeter = 21 centimeters

y = ___
Answer:
The length of the unknown side is 7 cm.

Explanation:
The perimeter of the figure is 21 cm,
and the lengths of the sides is 4 cm, 1 cm, 9 cm, y cm
so the perimeter of the figure is
p = 4 cm+ 1 cm+ 9 cm+ y cm
21 = 14+ y
y = 21 – 14
y = 7
=  7 cm.
The length of the unknown side is 7 cm.

Question 4.
Perimeter = 41 meters

y = ___
Answer:
The length of the unknown side is 8 m.

Explanation:
The perimeter of the figure is 41 m,
and the lengths of the sides is 3 m, 12 m, 10 m, 8 m, y m
so the perimeter of the figure is
p = 3+12+10+8+y
41 = 33 + y
y = 41-33
y = 8
=  8 m.
The length of the unknown side is 8 m.

Question 5.
Perimeter = 12 feet

d = ___
Answer:
The length of the sides of the triangle is 4 ft.

Explanation:
The perimeter of the triangle is 12 feet,
and the length of the side of the triangle is d ft,
so the perimeter of the triangle is
p = 3 a
12 = 3 × d
d = 12/3
= 4 ft.
The length of the sides of the triangle is 4 ft.

Question 6.
Perimeter = 50 inches

k = ___
Answer:
The length of the sides of the Hexagon is 4 in.

Explanation:
The perimeter of the Hexagon is 50 inches,
and the length of the side of the Hexagon is k in,
so the perimeter of the hexagon is
p = 5 a
50 = 5 × k
k = 50/5
= 10 in.
The length of the sides of the triangle is 10 in.

Question 7.
DIG DEEPER!
Each polygon has equal side lengths that are whole numbers. Which polygon could have a perimeter of 16 centimeters? Explain.

Answer:
The length of the sides of the octagon is 2 cm.

Explanation:
In the above three polygons, the second figure is an octagon, which has eight sides.
and the perimeter of the octagon is
p = 8 a
16 cm = 8 a
a = 16 /8
= 2 cm.

Question 8.
Number Sense
The area of a square is 25 square inches. What is its perimeter?
Answer:
The perimeter of the square is 20 inches.

Explanation:
The area of the square is 25 square inches, so
area = s^2
25 = s^2
s= 5 inches
so the perimeter of the square is
p = 4s
= 4×5
= 20 inches.

Question 9.
Modeling Real Life
The perimeter of the rectangular side walk is 260 meters. What are the lengths of the other three sides?

Answer:
The length of the other three sides is 10 m,10m,120 m.

Explanation:
The perimeter of the rectangular side walk is 260 meters,
and the length of the one side of the side walk is 120 m,
so the perimeter of the rectangular side walk is
p = 2( length + breadth)
260 = 2 ( 120 + breadth)
260/2 = 120 + breadth
130 = 120 + breadth
breadth = 130 – 120
= 10 m.
The length of the other three sides is 10 m,10m,120 m.

Question 10
Modeling Real Life
Two rectangular tables are pushed together. The perimeter is 40 feet. What are the lengths of the other three sides?

Answer:
The other length of the side of the small rectangle is 3 ft.
The other length of the side of the big rectangle is 5 ft.

Explanation:
Given the perimeter of the apartment is 40 feet,
first, we will take the big rectangle,
as the opposite sides of the rectangle are equal
and the length of the big rectangle is 5 ft
so the other length is also 5 ft.
as the perimeter of the small rectangle is 10 ft
and the length of the one side of the rectangle is 2 ft
so the other length of the small rectangle is
p = 2(length+breadth)
10 = 2(2 + breadth)
10/2 = 2+ breadth
5= 2 + breadth
breadth= 5 – 2
= 3 ft.
The other length of the side is 3 ft.

Review & Refresh

Write the time. Write another way to say the time.

Question 11.

Answer:
06: 48

Explanation:
Another way to say time is 06: 48

Question 12.

Answer:
03: 24

Explanation:
Another way to say time is 03: 24

Question 13.

Answer:

Explanation:
Another way to say the time is 11: 48

Lesson 15.4 Same Perimeter, Different Areas

Use color tiles to create two different rectangles that each have a perimeter of 16 units. Then draw your rectangles and label their dimensions. Do the rectangles have the same area? Explain how you know.

Answer:
No, the area of rectangle 1 and rectangle 2 is not the same.

Explanation:

Given the perimeter of the rectangle is 16 units
The length of rectangle 1 is 5 units
and the breadth of rectangle 1 is 3 units
so the area of rectangle 1 is
area = length × breadth
= 5×3
= 15 square units.
The length of rectangle 2 is 6 units
and the breadth of rectangle 2 is 2 units
so the area of rectangle 2 is
area = length × breadth
= 6×2
= 12 square units.
No, the area of rectangle 1 and rectangle 2 is not the same.

Repeated Reasoning
Draw another rectangle that has the same perimeter but different dimensions. Compare the area of the new rectangle to the rectangles above. What do you notice?
Answer:
No, the area of rectangle 1 and rectangle 2 is not the same.

Explanation:

The length of rectangle 1 is 4 units
and the breadth of rectangle 1 is 3 units
the perimeter of rectangle 1 is
p= 2(length+breadth)
= 2(4+3)
= 2(7)
= 14 units.
so the area of rectangle 1 is
area = length × breadth
= 4×3
= 12 square units.
The length of rectangle 2 is 5 units
and the breadth of rectangle 2 is 2 units
the perimeter of rectangle 1 is
p= 2(length+breadth)
= 2(5+2)
= 2(7)
= 14 units.
so the area of rectangle 2 is
area = length × breadth
= 5×2
= 10 square units.
No, the area of rectangle 1 and rectangle 2 is not the same.

Think and Grow : Same Perimeter, Different Areas

Example :
Find the perimeter and the area of Rectangle A. Draw a different rectangle that has the same perimeter. Which rectangle has the greater area?


Rectangle ___ has a greater area.
Answer:
The perimeter of the rectangle A is 20 m
and the area of the rectangle A is 24 m²
The perimeter of rectangle B is 20 m
and the area of the rectangle B is 16 m²
The rectangle A has a greater area.

Explanation:
Given the length of the rectangle is 6m
and the breadth of the rectangle is 4m,
so the perimeter of the rectangle is
p = 2( length + breadth)
= 2(6 + 4)
= 2(10)
= 20 m.
And the area of the rectangle is
a = length × breadth
= 6 × 4
= 24 m²

we can see in the above figure another rectangle with a length of 8m,
and the breadth is 2m,
so the perimeter of the rectangle is
p = 2(length + breadth)
= 2(8 + 2)
= 2(10)
= 20 m.
And the area of the rectangle is
area = length × breadth
= 8×2
= 16 m²
So the rectangle A has a greater area.

Show and Grow

Question 1.
Find the perimeter and area of Rectangle A. Draw a different rectangle that has the same perimeter. Which rectangle has the greater area?

Perimeter = ___
Area = ___

Perimeter = ___
Area = ___
Rectangle ___ has the greater area.
Answer:The perimeter of the rectangle A is 14 in
and the area of the rectangle A is 10 in²
The perimeter of rectangle B is 14 in
and the area of the rectangle B is 12 in²
The rectangle B has a greater area.

Explanation:
Given the length of the rectangle is 5 in
and the breadth of the rectangle is 2 in,
so the perimeter of the rectangle is
p = 2( length + breadth)
= 2(5 + 2)
= 2(7)
= 14 in.
And the area of the rectangle is
a = length × breadth
= 5×2
= 10 in²

we can see in the above figure another rectangle with a length of 4 in,
and the breadth is 3 in,
so the perimeter of the rectangle is
p = 2(length + breadth)
= 2(4 + 3)
= 2(7)
= 14 in.
And the area of the rectangle is
area = length × breadth
= 4×3
= 12 in²
So the rectangle B has a greater area.

Apply and Grow: Practice

Find the perimeter and area of Rectangle A. Draw a different rectangle that has the same perimeter. Which rectangle has the greater area?

Question 2.
Rectangle A

Perimeter = ___
Area = ___

Perimeter = ___
Area = ___
Rectangle ___ has the greater area.
Answer:
The perimeter of the rectangle A is 22 cm
and the area of the rectangle A is 11 cm²
The perimeter of rectangle B is 22 cm
and the area of the rectangle B is 30 cm²
The rectangle B has a greater area.

Explanation:
Given the length of the rectangle is 10 cm,
and the breadth of the rectangle is 1 cm,
so the perimeter of the rectangle is
p = 2( length + breadth)
= 2(10 + 1)
= 2(11)
= 22 cm.
And the area of the rectangle is
a = length × breadth
= 11 × 1
= 11 cm²

we can see in the above figure another rectangle with a length of 6 cm,
and the breadth is 5 cm,
so the perimeter of the rectangle is
p = 2(length + breadth)
= 2(6 + 5)
= 2(11)
= 22 cm.
And the area of the rectangle is
area = length × breadth
= 6×5
= 30 cm²
So the rectangle B has a greater area.

Question 3.
Rectangle A

Perimeter = ___
Area = ___

Rectangle B

Perimeter = ___
Area = ___
Rectangle ___ has the greater area
Answer:
The perimeter of the rectangle A is 20 m
and the area of the rectangle A is 21 m²
The perimeter of rectangle B is 20 m
and the area of the rectangle B is 24 m²
The rectangle B has a greater area.

Explanation:
Given the length of the rectangle is 7 m,
and the breadth of the rectangle is 3 m,
so the perimeter of the rectangle is
p = 2( length + breadth)
= 2(7 + 3)
= 2(10)
= 20 m.
And the area of the rectangle is
a = length × breadth
= 7 × 3
= 21 m²

we can see in the above figure another rectangle with a length of 6 m,
and the breadth is 4 m,
so the perimeter of the rectangle is
p = 2(length + breadth)
= 2(6 + 4)
= 2(10)
= 20 m.
And the area of the rectangle is
area = length × breadth
= 6×4
= 24 m²
So the rectangle B has a greater area.

Question 4.
MP Structure
Draw a rectangle that has the same perimeter as the one shown, but with a lesser area. What is the area ?

Answer:
The perimeter of the rectangle A is 26 ft
and the area of the rectangle A is 40 ft²
The perimeter of rectangle B is 26 ft
and the area of the rectangle B is 30 ft²
The rectangle B has a greater area.

Explanation:
Given the length of the rectangle is 5 ft,
and the breadth of the rectangle is 8 ft,
so the perimeter of the rectangle is
p = 2( length + breadth)
= 2(5 + 8)
= 2(13)
= 26 ft.
And the area of the rectangle is
a = length × breadth
= 5 × 8
= 40 ft

we can see in the above figure another rectangle with a length of 10 ft,
and the breadth is 3 ft,
so the perimeter of the rectangle is
p = 2(length + breadth)
= 2(10 + 3)
= 2(13)
= 26 ft.
And the area of the rectangle is
area = length × breadth
= 10×3
= 30 ft²
So the rectangle A has a greater area.

Think and Grow: Modeling Real Life

A paleontologist has 12 meters of twine to rope off a rectangular section of the ground. How long and wide should she make the roped-off section so it has the greatest possible area?
Draw to show:
She should make the roped-off section ___ meters long and ___ meters wide.
Answer:
She should make the roped-off section 4 meters long and 2 meters wide.

Explanation:
Given that a paleontologist has 12 meters of twine to rope off a rectangular section,
so if we take the length as 4 m and width as 2 m then we can get the greatest possible area,
so the area of the rectangular section is
area = length × breadth
= 4 m ×2 m
= 8 m²

Show and Grow

Question 5.
Newton has 16 feet of wood to make a rectangular sandbox. How long and wide should he make the sandbox so it has the greatest possible area?

Answer:
The greatest possible area of the rectangular sandbox is 15 ft²

Explanation:
As Newton has 16 feet of wood to make a rectangular sandbox,
so let the length be 5 ft and the wide be 3 ft to get the greatest possible area,
so the area of the rectangular sandbox is
area = length × breadth
= 5 ft × 3 ft
= 15 ft²
The greatest possible area of the rectangular sandbox is 15 ft²

Question 6.
DIG DEEPER!
You and Newton are building forts. You each have the same length of rope to make a rectangular perimeter for the forton the ground. Your roped-off section is shown. Newton’s section has a greater area than yours. Draw one way Newton could rope off his fort.

Descartes also builds a fort. He has the same length of rope as you to make a perimeter around his fort. Descartes’s roped-off section has a lesser area than yours. Draw one way Descartes could rope off his fort.

Answer:
Refer the below for detailed explanation.

Explanation:
The length of the rope is 7 ft
and the breadth of the rope is 3 ft
the perimeter is
p = 2(length+breadth)
= 2(7+3)
= 2(10)
= 20 ft.
The area of the rectangle is
area= length×breadth
= 7×3
= 21 square feet.

Let the length of the Newton’s rope is 6 ft
and the breadth of the Newton’s rope is 4 ft
the perimeter is
p = 2(length+breadth)
= 2(6+4)
= 2(10)
= 20 ft.
The area of the rectangle is
area= length×breadth
= 6×4
= 24 square feet.
And here Newton’s area is greater.

Let the length of the Descarte’s rope is 8 ft
and the breadth of the Descarte’s rope is 2 ft
the perimeter is
p = 2(length+breadth)
= 2(8+2)
= 2(10)
= 20 ft.
The area of the rectangle is
area= length×breadth
= 8×2
= 16 square feet.
And here Descartes area is lesser.

Same Perimeter, Different Areas Homework & Practice 15.4

Question 1.
Find the perimeter and the area of Rectangle A. Draw a different rectangle that has the same perimeter? Which rectangle has the greater area?

Perimeter = ___
Area = ___

Perimeter = ___
Area = ___
Rectangle __ has the greater area.
Answer:
The perimeter of the rectangle A is 7 cm
and the area of the rectangle A is 24 cm²
The perimeter of rectangle B is 26 ft
and the area of the rectangle B is 30 ft2
The rectangle B has a greater area.

Explanation:
Given the length of the rectangle is 7 cm,
and the breadth of the rectangle is 5 cm,
so the perimeter of the rectangle is
p = 2( length + breadth)
= 2(7 + 5)
= 2(12)
= 24 cm.
And the area of the rectangle is
a = length × breadth
= 7 × 5
= 35 cm².

we can see in the above figure rectangle with a length of 6.5 cm,
and the breadth is 5.5 cm,
so the perimeter of the rectangle is
p = 2(length + breadth)
= 2(6.5 + 5.5)
= 2(12)
= 24 ft.
And the area of the rectangle is
area = length × breadth
= 6.5×5.5
= 35.75 square feet
So the rectangle B has a greater area.

Question 2.
Patterns
Complete the pattern. Find the area of each rectangle.

Each rectangle has the same perimeter. As the area increases, what do you notice about the shape of the rectangle?
Answer:
As the area increases the shape of the figure was changed, we can see the figure was changed from rectangle to square.

Explanation:
As we know that the perimeter of the above figure is the same,
so the perimeter of the above figures is
p = 2 (length + breadth)
= 2 (1 m+9 m)
= 2(10 m)
= 20 m.
So, the perimeter of the above figures is 20 m.
The area of figure 1 is
area = length × breadth
= 1 m × 9 m
= 9 m².
The area of figure 2 is
= 8m × 2m
= 16 m².
The area of figure 3 is
= 7m × 3m
= 21 m².
Let the length of figure 4 be 6m and the breadth be 4m,
The area of figure 4 is
= 6m × 4m
= 24 m².
Let the length of figure 5 be 5m and the breadth be 5m,
The area of figure 5 is
= 5m × 5m
= 25 m².

Question 3.
Modeling Real Life
You are making a card with a 36-centimeter ribbon border. How long and wide should you make the card so you have the greatest possible area to write?

Answer:
The length and the breadth of the card is 9 cm.

Explanation:
Given the perimeter of the card with a ribbon border is 36 cm
so the length of the card is
p =4a
36 = 4a
a= 36/4
= 9 cm.
The length and the breadth of the card is 9 cm.
The area of the card is
a = length×breadth
= 9×9
= 81 square cm.

Question 4.
DIG DEEPER!
A school has two rectangular playgrounds that each have the same perimeter. The first playground is shown. The second has a lesser area than the first. Draw one way the second playground could look.

The school builds another playground. It has the same perimeter as the first. The third playground has a greater area than the first. Draw one way the third playground could look
Answer:

Explanation:

Review & Refresh

Question 5.
2 × 30 = ___
Answer:
60

Explanation:
On multiplying 2 × 30 we will get 60.

Question 6.
6 × 20 = ___
Answer:
120

Explanation:
On multiplying 6 × 20 we will get 120.

Question 7.
3 × 90 = ___
Answer:
270

Explanation:
On multiplying 3 × 90 we will get 270.

Lesson 15.5 Same Area, Different Perimeters

Explore and Grow

Use color tiles to create two different rectangles that each have an area of 18 square units. Then draw your rectangles and label their dimensions. Do the rectangles have the same perimeter? Explain how you know

Answer:
By comparing the perimeters of both rectangles, we can see that the perimeters are not the same.

Explanation:

In the above figure, we can see two colored rectangles
the length of rectangle 1 is 6 units
and the breadth of rectangle 1 is 3 units
so the perimeter of rectangle 1 is
p = 2(length+breadth)
= 2(6+3)
= 2(9)
= 18 units
and the area of rectangle 1 is
area= length×breadth
= 6×3
= 18 square units
The length of rectangle 2 is 9 units
and the breadth of rectangle 2 is 2 units
so the perimeter of rectangle 2 is
p = 2(length+breadth)
= 2(9+2)
= 2(11)
= 22 units.
and the area of rectangle 2 is
area= length×breadth
= 9×2
= 18 square units.
By comparing the perimeters of both rectangles, we can see that the perimeters are not the same.

Repeated Reasoning
As the perimeter increases and the area stays the same, what do you notice about the shape of the rectangle?

Think and Grow : Same Area, Different Perimeters

Example
Find the area and the perimeter of Rectangle A. Draw a different rectangle that has the same area. Which rectangle has the lesser perimeter?

Area = 2 × 6
= _____
Perimeter = 6 + 2 + 6 + 2
= ______

Area = ___ × ___
= ____
Perimeter = ___ + ___ + ___ + ___
= ____
Rectangle ___ has the lesser perimeter.
Answer:
The perimeter of the rectangle A is 16 ft
and the area of the rectangle A is 12 ft²
The perimeter of rectangle B is 14 ft
and the area of the rectangle B is 12 ft²
The rectangle A has a lesser area.

Explanation:
Given the length of the rectangle is 6 ft,
and the breadth of the rectangle is 2 ft,
so the perimeter of the rectangle is
p = 2( length + breadth)
= 2(6 + 2)
= 2(8)
= 16 ft.
And the area of the rectangle is
a = length × breadth
= 6 × 2
= 12 ft²

we can see in the above figure rectangle B and the length be 4 ft,
and the breadth be 3ft,
so the perimeter of the rectangle is
p = 2(length + breadth)
= 2(4 + 3)
= 2(7)
= 14 ft.
And the area of the rectangle is
area = length × breadth
= 4×3
= 12 ft²
So the rectangle A has a lesser area.

Show and Grow

Question 1.
Find the area and the perimeter of Rectangle A. Draw a different rectangle that has the same area. Which rectangle has the lesser perimeter?

Area = ___
Perimeter = ___

Area = ___
Perimeter = ___
Rectangle __ has the lesser perimeter.
Answer:
The perimeter of the rectangle A is 16 ft
and the area of the rectangle A is 12 ft²
The perimeter of rectangle B is 14 ft
and the area of the rectangle B is 12 ft²
The rectangle A has a lesser area.

Explanation:
Given the length of the rectangle is 6 cm,
and the breadth of the rectangle is 6 cm,
so the perimeter of the rectangle is
p = 2( length + breadth)
= 2(6 + 6)
= 2(12)
= 24 cm.
And the area of the rectangle is
a = length × breadth
= 6 × 6
= 36 cm²

we can see in the above figure rectangle B and the length be 9 cm,
and the breadth be 4 cm,
so the perimeter of the rectangle is
p = 2(length + breadth)
= 2(9 + 4)
= 2(13)
= 26 cm.
And the area of the rectangle is
area = length × breadth
= 9×4
= 36 cm²
So the rectangle A has a lesser perimeter.

Apply and Grow: Practice

Find the area and the perimeter of Rectangle A. Drawa different rectangle that has the same area. Which rectangle has the lesser perimeter?
Answer:

Question 2.
Rectangle A

Area = ___
Perimeter = ___
Rectangle B

Area = ___
Perimeter = ___
Rectangle ___ has the lesser perimeter.
Answer:
The perimeter of the rectangle A is 24 in
and the area of the rectangle A is 20 in²
The perimeter of rectangle B is 18 in
and the area of the rectangle B is 20 in²
The rectangle B has a lesser perimeter.

Explanation:
Given the length of the rectangle is 10 in,
and the breadth of the rectangle is 2 in,
so the perimeter of the rectangle is
p = 2( length + breadth)
= 2(10 + 2)
= 2(12)
= 24 in.
And the area of the rectangle is
a = length × breadth
= 10 × 2
= 20 in²

we can see in the above figure rectangle B and the length be 5 in,
and the breadth be 4 in,
so the perimeter of the rectangle is
p = 2(length + breadth)
= 2(5 + 4)
= 2(9)
= 18 in.
And the area of the rectangle is
area = length × breadth
= 5×4
= 20 in²
So the rectangle B has a lesser perimeter.

Question 3.
Rectangle A

Area = ___
Perimeter = ___
Rectangle B

Area = ___
Perimeter = ___
Rectangle ___ has the lesser perimeter.
Answer:
The perimeter of the rectangle A is 12 m
and the area of the rectangle A is 8 m²
The perimeter of rectangle B is 18 m
and the area of the rectangle B is 8 m²
The rectangle A has a lesser perimeter.

Explanation:
Given the length of the rectangle is 4 m,
and the breadth of the rectangle is 2 m,
so the perimeter of the rectangle is
p = 2( length + breadth)
= 2(4 + 2)
= 2(6)
= 12 m.
And the area of the rectangle is
a = length × breadth
= 4 × 2
= 8 m²

we can see in the above figure rectangle B and the length be 8 m,
and the breadth be 1 m,
so the perimeter of the rectangle is
p = 2(length + breadth)
= 2(8+ 1)
= 2(9)
= 18 m.
And the area of the rectangle is
area = length × breadth
= 8×1
= 8 m²
So the rectangle A has a lesser perimeter.

Question 4.
DIG DEEPER!
The perimeter of a blue rectangle is 10 feet. The perimeter of a green rectangle is 14 feet. Both rectangles have the same area. Find the area and the dimensions of each rectangle.
Answer:

Explanation:
Given the perimeter of the blue rectangle is 10 ft and
the perimeter of the green rectangle is 14 ft

Think and Grow: Modeling Real Life

You have 40 square patio bricks that are each 1 foot long and 1 foot wide. You want to make a rectangular patio with all of the bricks. How long and wide should you make the patio so it has the least possible perimeter?

Draw to show:
You should make the patio ___ feet long and ___ feet wide.
Answer:
You should make the patio 8 feet long and 5 feet wide.
The least possible perimeter is 26 feet.

Explanation:

As there are 40 square patio bricks and each brick is 1 foot long and 1 foot wide
so to make a rectangular patio we need
the length of the rectangular patio be 8 feet
and the breadth of the rectangular patio be 5 feet
so the perimeter of the rectangular patio is
p= 2(length+breadth)
= 2(8+5)
= 2(13)
= 26 feet.
The least possible perimeter is 26 feet.

Show and Grow

Question 5.
Your friend has 16 square foam tiles that are each 1 foot long and 1 foot wide. He wants to make a rectangular exercise space with all of the tiles. How long and wide should he make the exercise space so it has the least possible perimeter?

Answer:
The least possible perimeter is 16 feet.

Explanation:
As there are 16 square foam tiles and each foam tile is 1 foot long and 1 foot wide
so to make an exercise space we need
the length of the exercise space be 4 feet
and the breadth of the exercise space be 4 feet
so the perimeter of the exercise space is
p= 2(length+breadth)
= 2(4+4)
= 2(8)
= 16 feet.
The least possible perimeter is 16 feet.

Question 6.
DIG DEEPER!
You and your friend each use fencing to make a rectangular playpen for a puppy. Each pen has the same area. Your pen is shown. Your friend’s pen uses less fencing than yours. Draw one way your friend could make her pen.

Your cousin makes a playpen for a puppy. His pen has the same area as your pen. Your cousin’s pen uses more fencing than yours. Draw one way your cousin could make his pen.
Answer:

Same Area, Different Perimeters Homework & Practice 15.5

Question 1.
Find the area and the perimeter of Rectangle A. Drawa different rectangle that has the same area. Which rectangle has the lesser perimeter?

Area = ___
Perimeter = ___

Area = ___
Perimeter = ___
Rectangle __ has the lesser perimeter.
Answer:
The perimeter of rectangle A is 4 in
and the area of rectangle A is 20 square inches.
It is not possible to draw a rectangle that has the same area and different perimeter.

Explanation:
Given the length of the rectangle is 4 in,
and the breadth of the rectangle is 4 in,
so the perimeter of the rectangle is
p = 2( length + breadth)
= 2(4+4)
= 2(8)
= 16 in.
And the area of the rectangle is
a = length × breadth
= 4 × 4
= 16 square inches.

Question 2.
Structure
The dimensions of a rectangle are 4 feet by 10 feet. Which shape has the same area, but a different perimeter?

Answer:
The yellow shape rectangle has the same area and different perimeter.

Explanation:
Given the dimensions of the rectangle are 4 feet by 10 feet
so the perimeter of the rectangle is
p= 2(length+breadth)
= 2(4+10)
= 2(14)
= 28 feet.
The area of the rectangle is
area = length×breadth
= 10×4
= 40 square feet.
Here, we can see the yellow rectangle has a length of 8 feet
and the breadth of the rectangle is 5 feet
so the perimeter of the rectangle is
p = 2(length+beadth)
= 2(8+5)
= 2(13)
= 26 feet.
and the area of the rectangle is
area = length×breadth
= 8×5
= 40 square feet.

Question 3.
MP Reasoning
The two fields have the same area. Players run one lap around each field. At which field do the players run farther?

Answer:
As the perimeter of field A is greater than field B so in field A the players run father.

Explanation:
Let the length of field A is 10m
and the breadth of field A is 2m
so the perimeter of field A is
p = 2(length+breadth)
= 2(10+2)
= 2(12)
= 24 m.
and the area of field A is
area= length×breadth
= 10×2
= 20 square meters.
Let the length of field B is 5m
and the breadth of field B is 4m
so the perimeter of field B is
p = 2(length+breadth)
= 2(5+4)
= 2(9)
= 18 m.
and the area of field A is
area= length×breadth
= 5×4
= 20 square meters.
As the perimeter of field A is greater than field B so in field A the players run father.

Question 4.
Modeling Real Life
You have 24 square pieces of T-shirt that are each 1 foot long and 1 foot wide. You want to make a rectangular T-shirt quilt with all of the pieces. How long and wide should you make the quilt so it has the least possible perimeter?

Answer:
The least possible perimeter is 20 feet.

Explanation:
As there are 24 square pieces of T-shirt and each was 1 foot long and 1 foot wide
so to make a rectangular T-shirt quilt with all of the pieces we need
the length of the rectangular T-shirt quilt be 6 feet
and the breadth of the rectangular T-shirt quilt be 4 feet
so the perimeter of the rectangular T-shirt quilt is
p= 2(length+breadth)
= 2(6+4)
= 2(10)
= 20 feet.
The least possible perimeter is 20 feet.

Question 5.
DIG DEEPER!
You and Descartes each have40 cobblestone tiles to arrange in to a rectangular pathway. Your pathway is shown. Descartes’s pathway has a lesser perimeter than yours. Draw one way Descartes could make his pathway.

Newton also makes a rectangular pathway with 40 cobblestone tiles. His pathway has a greater perimeter than yours. Draw one way Newton could make his pathway.
Answer:

Review & Refresh

Identify the number of right angles and pairs of parallel sides.

Question 6.

Right angles: ___
Pairs of Parallel sides: ___
Answer:
Right angles: 1
Pairs of Parallel sides: 2.

Explanation:
In the above figure, we can see there are the right angle is 1, and the pairs of parallel sides are 2.

Question 7.

Right angles: ___
Pairs of parallel sides: ____
Answer:
Right angles: 4.
Pairs of Parallel sides: 2.

Explanation:
In the above figure, we can see there are the right angle is 4, and the pairs of parallel sides are 2.

Find Perimeter and Area Performance Task

You and your cousin build a tree house.

Question 1.
The floor of the tree house is in the shape of a quadrilateral with parallel sides that are 4 feet long and 10 feet long. The other 2 sides are equal in length. The perimeter is 24 feet. Sketch the floor and label all of the side lengths.

Answer:
The length of the sides is 6 feet.

Explanation:
The perimeter of the floor is 24 feet
so the length of the floor be 6 feet
as the two other sides are also equal
so the other side length also be 6 feet
and let’s check the perimeter
p = 2(length+breadth)
= 2(6+6)
= 2(12)
= 24 feet.

Question 2.
Each rectangular wall of the tree house is 5 feet tall. How many square feet of wood is needed for all of the walls?
Answer:

Explanation:

Question 3.
You cut out a door in the shape of a rectangle with sides that are whole numbers. Its area is 8 square feet. What is the height of the door?
Answer:
The height of the rectangular door is 4 feet.

Explanation:
The area of the rectangular shape door is 8 square feet
as the sides of the rectangular door are whole numbers
so the length rectangular door be 4 feet
and the breadth be 2 feet
then we can get the area 8 square feet
let’s check the area
area = length×breadth
= 4×2
= 8 square feet.
So the height of the rectangular door is 4 feet.

Question 4.
You want to paint the floor and walls on the inside of your tree house. The area of the floor is 28 square feet. Each quart of paint covers 100 square feet.
a. How many quarts of paint do you need to buy?
b. Do you have enough paint to paint the outside walls of the tree house? Explain.

Answer:
a. 2,800 square feet quarts of paint we need to buy.

Explanation:
a. The area of the floor is 28 square feet and each quart of paint covers 100 square feet, so we need to buy
28×100= 2,800 square feet quarts of paint.

b.

Find Perimeter and Area Activity

Perimeter Roll and Conquer

Directions:
1. Players take turns rolling two dice.
2. On your turn, draw a rectangle on the board using the numbers on the dice as the side lengths. Your rectangle cannot cover another rectangle.
3. Write an equation tofind the perimeter of the rectangle.
4. If you cannot fit a rectangle on the board, then you lose your turn. Play 10 rounds, if possible.
5. Add all of your rectangles’ perimeters together. The player with the greatest sum wins!

Answer:

Explanation:

Find Perimeter and Area Chapter Practice

15.1 Understand Perimeter

Find the perimeter of the figure

Question 1.

Perimeter = ___
Answer:
The perimeter of the rectangle is 18 cm.

Explanation:
In the above figure, we can see the rectangle
with a length of 5 cm,
and the breadth of 4 cm
the perimeter of the rectangle is
p = 2 (length + breadth)
= 2 (5+4)
= 2(9)
= 18 cm.

Question 2.

Perimeter = ___
Answer:
The perimeter of the figure is 26 ft.

Explanation:
To find the perimeter of the above figure,
we will add the lengths of all sides of the figure
the sides of the above figure is 2 ft, 8 ft, 3 ft, 2 ft, 2 ft, 2 ft, 1 ft, 1 ft, 2 ft, 3 ft
the perimeter of the above figure is
p = 2+8+3+2+2+2+1+1+2+3
= 26 ft.

Question 3.
Draw a figure that has a perimeter of 10 inches.

Answer:

15.2 Find Perimeter of Polygons

Find the perimeter of the polygon

Question 4.

Perimeter = ___
Answer:
The perimeter of the polygon is 33 cm.

Explanation:
To find the perimeter of the polygon, we will add all the sides of the polygon
so the sides of the polygon are 9 cm, 6 cm, 8 cm, 10 cm
the perimeter of the polygon is
p = 9 cm +6 cm +8 cm +10 cm
= 33 cm.

Question 5.

Perimeter = ___
Answer:
The perimeter of the figure is 27 ft.

Explanation:
To find the perimeter of the figure, we will add all the sides of the figure
so the sides of the perimeter is 5 ft, 11 ft, 7 ft, 3 ft, 1 ft
the perimeter of the figure is
p = 5 ft+11 ft+ 7 ft+3 ft+1 ft
= 27 ft.

Question 6.
Parallelogram

Perimeter = ___
Answer:
The perimeter of the parallelogram is 12 m.

Explanation:
Given the length of the parallelogram is 4 m
and the breadth of the parallelogram is 2 m
the perimeter of the parallelogram is
p = 2 (length + breadth)
= 2( 4 m+ 2 m)
= 2(6 m)
= 12 m

Find the perimeter of the polygon

Question 7.
Rhombus

Perimeter = ___
Answer:
The perimeter of the rhombus is 36 cm.

Explanation:
Given the length of the side of the rhombus is 9 cm
and the perimeter of the rhombus is
p = 4a
= 4× 9 cm
= 36 cm.

Question 8.
Rectangle

Perimeter = ___
Answer:
The perimeter of the rectangle is 26 in.

Explanation:
The length of the rectangle is 5 inch
and the breadth of the rectangle is 8  inch
the perimeter of the rectangle is
p = 2( length + breadth)
= 2( 5 in+ 8 in)
= 2(13 in)
= 26 in.
So the perimeter of the rectangle is 26 in.

Question 9.
Square

Perimeter = ___
Answer:
The perimeter of the square is 28 ft.

Explanation:
The length of the square is 7 ft
so the perimeter of the square is
perimeter= 4a
= 4×7
= 28 ft.

Question 10.
Modeling Real Life 
You want to put lace around the tops of the two rectangular lampshades. How many centimeters of lace do you need?

Answer:
We need 1,120 square centimeters.

Explanation:
The length of the rectangular lampshades is 35 cm
The breadth of the rectangular lampshades is 32 cm
and the area of the rectangular lampshades is
area= length×breadth
= 32×35
= 1,120 square cm.
So 1,120 square centimeters of lace you need.

15.3 Find Unknown Side Lengths

Find the unknown side length.

Question 11.
Perimeter = 22 feet

d = ____
Answer:
The length of the other side is 9 ft.

Explanation:
Given the perimeter of the above figure is 22 feet
and the length of the sides of the figure is 6 ft, 7 ft, d ft
so the perimeter of the figure is
p = 6 ft+ 7 ft+ d ft
22  ft = 13 ft + d ft
d = 22 ft – 13 ft
= 9 ft.
So, the length of the other side is 9 ft.

Question 12.
Perimeter = 31 inches

k = ___
Answer:
The length of the other side is 5 in.

Explanation:
Given the perimeter of the above figure is 31 inches
and the length of the sides of the figure is 10 in, 4 in, 12 in and k in.
so the perimeter of the figure is
p = 10 in+ 4 in+ 12 in+k in
31 in = 26 in + k in
k = 31 in – 26 in
= 5 in.
So, the length of the other side is 5 in.

Question 13.
Perimeter = 34 meters

y = ___
Answer:
The length of the other side is 7 m.

Explanation:
Given the perimeter of the above figure is 34 meters
and the length of the sides of the figure is 11 m, 8 m, 2 m, 1 m, 5 m, y m.
so the perimeter of the figure is
p = 11 m+ 8 m+ 2 m+ 1 m+ 5 m+ y m
34 m = 27 m + y m
y = 34 m – 27 m
= 7 m.
So, the length of the other side is 7 m.

Find the unknown side length.

Question 14.
Perimeter = 24 feet

k = ___
Answer:
The length of the sides of the triangle is 8 feet.

Explanation:
The perimeter of the triangle is 24 feet
and the perimeter of the triangle is
p = 3a
24 feet = 3a
a= 24/3
= 8 feet.
So, the length of the sides of the triangle is 8 feet.

Question 15.
Perimeter = 16 meters

y = ___
Answer:
The length of the sides of the square is 4 meters.

Explanation:
The perimeter of the triangle is 16 meters
and the perimeter of the triangle is
p = 4a
16 meters = 4a
a= 16/4
= 4 meters.
So, the length of the sides of the triangle is 4 meters.

Question 16.
Perimeter = 30 inches

d = ___
Answer:
The length of the sides of the pentagon is 6 inches.

Explanation:
The perimeter of the pentagon is 30 inches
and the perimeter of the pentagon is
p = 5a
30 inches = 5a
d= 30/5
= 6 inches.
So, the length of the sides of the pentagon is 6 inches.

Question 17.

Number Sense
A rectangle has a perimeter of 38 centimeters. The left side length is 10 centimeters. What is the length of the top side?
Answer:
The length of the top side is 9 cm.

Explanation:
Given the perimeter of the rectangle is 38 cm and
the left side length is 10 cm
Let the length of the top side be X, so
perimeter of the rectangle is
p = 2( length +breadth)
38 = 2(10+X)
38/2 = 10 + X
19 = 10 + X
X = 19 – 10
= 9 cm.
So the length of the top side is 9 cm.

15.4 Same Perimeter, Different Area

Question 18.
Find the perimeter and area of Rectangle A. Drawa different rectangle that has the same perimeter. Which rectangle has the greater area?

Perimeter = ____
Area = ___

Perimeter = ____
Area = ___
Rectangle ___ has the greater area.
Answer:
The perimeter of the rectangle A is 14 m
The area of the rectangle A is 10 square meters
The perimeter of the rectangle B is 14 m
The area of the rectangle B is 12 square meters
The rectangle B has greater area.

Explanation:
The length of the rectangle is 5m
and the breadth of the rectangle is 2m
the perimeter of the rectangle is
p= 2(length+breadth)
= 2(5+2)
= 2(7)
= 14 m
and the area of the rectangle is
area = length×breadth
= 2×5
= 10 square meters.

In the above image, we can see the length of the rectangle is 4 m
and the breadth of the rectangle is 3 m
so the perimeter of the rectangle is
p= 2(length+breadth)
= 2(4+3)
= 2(7)
= 14 m.
and the area of the rectangle is
area = length×breadth
= 4×3
= 12 square meters.
The rectangle B has greater area.

Question 19.
Patterns
Each Rectangle has the same perimeter. Are the areas increasing or decreasing ? Explain.

Answer:
As we can see in the above images the length of the images was increasing one by one and the breadth is decreasing, so the areas increasing or decreasing will depend upon the breadth of the rectangle. So, if the breadth is also increasing then the area will also be increasing. And if the breadth was decreasing then the area will also be decreasing.

15.5 Same Area, Different Perimeters

Question 20.
Find the area and the perimeter of Rectangle A. Drawa different rectangle that has the same area. Which rectangle has the lesser perimeter?

Area = ___
Perimeter = ___

Area = ___
Perimeter = ___
Rectangle ___ has the lesser perimeter.
Answer:
The perimeter of the rectangle A is 14 m
The area of the rectangle A is 10 square meters
The perimeter of the rectangle B is 14 m
The area of the rectangle B is 12 square meters
The rectangle B has greater area.

Explanation:
The length of the rectangle is 10 in
and the breadth of the rectangle is 5 in
the perimeter of the rectangle is
p= 2(length+breadth)
= 2(10+5)
= 2(15)
= 30 in
and the area of the rectangle is
area = length×breadth
= 10×5
= 50 square inches.

Question 21.
Reasoning
The two dirt-bike parks have the same area. Kids ride dirt bikes around the outside of each park. At which park do the kids ride farther ? Explain.

Answer:
As the length of the park B is longer, so at the park B kids rid farther than the park A.

Find Perimeter and Area Cumulative practice 1 – 15

Question 1.
A mango has a mass that is 369 grams greater than the apple. What is the mass of the mango?
A. 471 grams
B. 369 grams
C. 267 grams
D. 461 grams

Answer:
B.

Explanation:
The mass of the mango is 369 grams greater than apple

Question 2.
Which term describes two of the shapes shown, but all three of the shapes?

A. polygon
B. rectangle
C. square
D. parallelogram
Answer:
B, C, D.

Explanation:
In the above figures, we can see the parallelogram and the square. As the square is also known as a rectangle so we will choose option B also.

Question 3.
A rectangular note card has an area of 35 square inches. The length of one of its sides is 7 inches. What is the perimeter of the note card?
A. 5 inches
B. 24 inches
C. 84 inches
D. 12 inches
Answer:
The breadth of the rectangular note card is 24 inches.

Explanation:
The area of the rectangular note card is 35 square inches and the length of one of its sides is 7 inches
so the breadth of the rectangular note card is
area = length × breadth
35 = 7 × breadth
breadth = 35/7
= 5 inches.
The perimeter of the rectangular note card is
p = 2(length+breadth)
= 2(7+5)
= 2(12)
= 24 inches.
The breadth of the rectangular note card is 24 inches.

Question 4.
How many minutes are equivalent to4 hours?
A. 400 minutes
B. 240 minutes
C. 24 minutes
D. 40 minutes
Answer:
B

Explanation:
The number of minutes is equivalent to 4 hours is
4× 60= 240 minutes.

Question 5.
A balloon artist has 108 balloons. He has 72 white balloons, and an equal number of red, blue, green, and purple balloons. How many purple balloons does he have?
A. 36
B. 180
C. 9
D. 32

Answer:
9 balloons.

Explanation:
As a balloon artist has 108 balloons and he has 72 white balloons
and the remaining balloons are 108 – 72= 36 balloons
and an equal number of red, blue, green, and purple balloons
which means 36 balloons are equally divided by 4 colors of balloons, so
36÷4 = 9 balloons.
So the purple balloons are 9.

Question 6.
Which statements about the figures are true?

Answer:
The shapes have different perimeters.
The shapes have the same area.

Explanation:
The length of the side of the square is 6 in,
and the perimeter of the square is
p = 4a
= 4×6 in
= 24 in
The area of the square is a^2
= 6 in×6 in
36 in^2.

Question 7.
The graph show many students ordered each lunch option.

Part A How many students ordered lunch?
Part B Choose a lesser value for the key. How will the graph change?
Answer:
Part A: 60 students ordered lunch.
Part B: Turkey hot dog has a lesser value.

Explanation:
Part A:
The number of students who ordered lunch is
the grilled chicken was ordered by 21 students
Turkey hot dog was ordered by 9 students
A peanut butter and jelly sandwich was ordered by 12 students
the salad bar was ordered by 18 students
so the number of students who ordered lunch is
21+9+12+18= 60 students.

Part B:
The turkey hot dog was ordered by 9 students which is a lesser value.

Question 8.
Find the sum

Answer:
935

Explanation:
The sum of the above given numbers is 935

Question 9.
What is the perimeter of the figure?

A. 26 units
B. 22 units
C. 20 units
D. 16 unit
Answer:
B

Explanation:
The sides of the figure is 2,4,1,2,1,1,2,2,1,1,2,1,1,1
and the perimeter of the figure is
p = 2+4+1+2+1+1+2+2+1+1+2+1+1+1
= 22 units.

Question 10.
Which bar graph correctly shows the data?

Answer:
Graph B.

Explanation:
Graph B shows the correct graph data.

Question 11.
Which polygons have at least one pair of parallel sides?

Answer:
Red color polygon.

Explanation:
The red color polygon has one pair of parallel sides, as it is a trapezoid.

Question 12.
The perimeter of the polygon is 50 yards. What is the missing side length?
A. 41 yards
B. 10 yards
C. 91 yards
D. 9 yards

Answer:
The missing side length is 9 yards.

Explanation:
Given the perimeter of the polygon is 50 yards
Let the missing side length be X yd
and the lengths of the sides of the polygon is 15 yds, 6 yds, 13 yds, 7 yds, and X yd,
So the perimeter of the polygon is
p = 15 yd+6 yd+13 yd+ 7 yd+ X yd
50 yards = 41 yards + X Yards
X = 9 yards.
So the missing side length is 9 yards.