4.2 Modeling with Linear Functions

Building a linear model from a verbal description, deciding what the slope and intercept mean in context, and using the result to predict and interpret.

Algebra and Trigonometry 2e, 4.2 Modeling with Linear Functions

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66 exercises

Using a Linear Model to Investigate a Town’s Population

A town’s population has been growing linearly. In 2004, the population was 6,200. By 2009, the population had grown to 8,100. Assume this trend continues.

ⓐPredict the population in 2013.
ⓑIdentify the year in which the population will reach 15,000.

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4.2 Exercise 7

A company sells doughnuts. They incur a fixed cost of $25,000 for rent, insurance, and other expenses. It costs $0.25 to produce each doughnut.

ⓐ Write a linear model to represent the cost $C$ of the company as a function of $x ,$ the number of doughnuts produced.
ⓑ Find and interpret the y-intercept.

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4.2 Exercise 8

A city’s population has been growing linearly. In 2008, the population was 28,200. By 2012, the population was 36,800. Assume this trend continues.

ⓐPredict the population in 2014.
ⓑIdentify the year in which the population will reach 54,000.

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4.2 Exercise 11

There is a straight road leading from the town of Timpson to Ashburn 60 miles east and 12 miles north. Partway down the road, it junctions with a second road, perpendicular to the first, leading to the town of Garrison. If the town of Garrison is located 22 miles directly east of the town of Timpson, how far is the road junction from Timpson?

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4.2 Exercise 13

Explain how to find the input variable in a word problem that uses a linear function.

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4.2 Exercise 15

Explain how to interpret the initial value in a word problem that uses a linear function.

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4.2 Exercise 17

Find the area of a parallelogram bounded by the y-axis, the line $x = 3 ,$ the line $f \left(x\right) = 1 + 2 x ,$ and the line parallel to $f \left(x\right)$ passing through $\left(\text{2} , \text{7}\right) .$

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4.2 Exercise 19

Find the area of a triangle bounded by the y-axis, the line $f \left(x\right) = 9 – \frac{6}{7} x ,$ and the line perpendicular to $f \left(x\right)$ that passes through the origin.

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4.2 Exercise 21

Consider this scenario: A town’s population has been decreasing at a constant rate. In 2010 the population was 5,900. By 2012 the population had dropped to 4,700. Assume this trend continues.

Predict the population in 2016.

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4.2 Exercise 23

Consider this scenario: A town’s population has been increased at a constant rate. In 2010 the population was 46,020. By 2012 the population had increased to 52,070. Assume this trend continues.

Predict the population in 2016.

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4.2 Exercise 25

Consider this scenario: A town has an initial population of 75,000. It grows at a constant rate of 2,500 per year for 5 years.

Find the linear function that models the town’s population $P$ as a function of the year, $t ,$ where $t$ is the number of years since the model began.

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4.2 Exercise 27

Consider this scenario: A town has an initial population of 75,000. It grows at a constant rate of 2,500 per year for 5 years.

If the function $P$ is graphed, find and interpret the x- and y-intercepts.

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4.2 Exercise 29

Consider this scenario: A town has an initial population of 75,000. It grows at a constant rate of 2,500 per year for 5 years.

When will the population reach 100,000?

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4.2 Exercise 31

Consider this scenario: The weight of a newborn is 7.5 pounds. The baby gained one-half pound a month for its first year.

Find the linear function that models the baby’s weight $W$ as a function of the age of the baby, in months, $t .$

2.5

4.2 Exercise 33

Consider this scenario: The weight of a newborn is 7.5 pounds. The baby gained one-half pound a month for its first year.

If the function $W$ is graphed, find and interpret the x- and y-intercepts.

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4.2 Exercise 35

Consider this scenario: The weight of a newborn is 7.5 pounds. The baby gained one-half pound a month for its first year.

When did the baby weight 10.4 pounds?

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4.2 Exercise 37

Consider this scenario: The number of people afflicted with the common cold in the winter months steadily decreased by 205 each year from 2005 until 2010. In 2005, 12,025 people were inflicted.

Find the linear function that models the number of people inflicted with the common cold $C$ as a function of the year, $t .$

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4.2 Exercise 39

Consider this scenario: The number of people afflicted with the common cold in the winter months steadily decreased by 205 each year from 2005 until 2010. In 2005, 12,025 people were inflicted.

If the function $C$ is graphed, find and interpret the x- and y-intercepts.

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4.2 Exercise 41

Consider this scenario: The number of people afflicted with the common cold in the winter months steadily decreased by 205 each year from 2005 until 2010. In 2005, 12,025 people were inflicted.

When will the number of people afflicted with the common cold reach 0?

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4.2 Exercise 43

Find the linear function $y ,$ where $y$ depends on $t ,$ the number of years since 1980.

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4.2 Exercise 45

Find and interpret the x-intercept.

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4.2 Exercise 47

Find the linear function $y ,$ where $y$ depends on $t ,$ the number of years since 1980.

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4.2 Exercise 49

Find and interpret the x-intercept.

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4.2 Exercise 51

In which state have home values increased at a higher rate?

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4.2 Exercise 53

If we assume the linear trend existed before 1950 and continues after 2000, the two states’ median house values will be (or were) equal in what year? (The answer might be absurd.)

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4.2 Exercise 55

If these trends were to continue, what would be the median home value in Indiana in 2010?

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4.2 Exercise 57

In 2004, a school population was 1001. By 2008 the population had grown to 1697. Assume the population is changing linearly.

ⓐHow much did the population grow between the year 2004 and 2008?
ⓑHow long did it take the population to grow from 1001 students to 1697 students?
ⓒWhat is the average population growth per year?
ⓓWhat was the population in the year 2000?
ⓔFind an equation for the populat…

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4.2 Exercise 59

A phone company has a monthly cellular plan where a customer pays a flat monthly fee and then a certain amount of money per minute used for voice or video calling. If a customer uses 410 minutes, the monthly cost will be $71.50. If the customer uses 720 minutes, the monthly cost will be $118.

ⓐFind a linear equation for the monthly cost of the cell plan as a function of x, the number of monthly m…

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4.2 Exercise 61

In 1990, the moose population in a park was measured to be 4,360. By 1999, the population was measured again to be 5,880. Assume the population continues to change linearly.

ⓐFind a formula for the moose population, P since 1991.
ⓑWhat does your model predict the moose population to be in 2003?

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4.2 Exercise 63

The Federal Helium Reserve held about 16 billion cubic feet of helium in 2010 and is being depleted by about 2.1 billion cubic feet each year.

ⓐ Give a linear equation for the remaining federal helium reserves, $R ,$ in terms of $t ,$ the number of years since 2010.
ⓑ In 2015, what will the helium reserves be?
ⓒ If the rate of depletion doesn’t change, in what year will the Federal Helium Reserve be deple…

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4.2 Exercise 65

You are choosing between two different prepaid cell phone plans. The first plan charges a rate of 26 cents per minute. The second plan charges a monthly fee of $19.95 plus 11 cents per minute. How many minutes would you have to use in a month in order for the second plan to be preferable?

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4.2 Exercise 67

When hired at a new job selling jewelry, you are given two pay options:

Option A: Base salary of $17,000 a year with a commission of 12% of your sales

Option B: Base salary of $20,000 a year with a commission of 5% of your sales

How much jewelry would you need to sell for option A to produce a larger income?

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4.2 Exercise 69

When hired at a new job selling electronics, you are given two pay options:

Option A: Base salary of $20,000 a year with a commission of 12% of your sales

Option B: Base salary of $26,000 a year with a commission of 3% of your sales

How much electronics would you need to sell for option A to produce a larger income?

2.5

Using a Linear Model to Investigate a Town’s Population

A town’s population has been growing linearly. In 2004 the population was 6,200. By 2009 the population had grown to 8,100. Assume this trend continues.

ⓐ Predict the population in 2013.
ⓑ Identify the year in which the population will reach 15,000.

2.5

2.3 Exercise 2

A company sells doughnuts. They incur a fixed cost of $25,000 for rent, insurance, and other expenses. It costs $0.25 to produce each doughnut.

ⓐ Write a linear model to represent the cost $C$ of the company as a function of $x ,$ the number of doughnuts produced.
ⓑ Find and interpret the y-intercept.

2.5

2.3 Exercise 3

A city’s population has been growing linearly. In 2008, the population was 28,200. By 2012, the population was 36,800. Assume this trend continues.

ⓐ Predict the population in 2014.
ⓑ Identify the year in which the population will reach 54,000.

2.5

2.3 Exercise 6

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2.3 Exercise 8

Explain how to find the input variable in a word problem that uses a linear function.

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2.3 Exercise 10

Explain how to interpret the initial value in a word problem that uses a linear function.

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2.3 Exercise 12

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2.3 Exercise 14

Find the area of a triangle bounded by the y-axis, the line $f \left(x\right) = 9 – \frac{6}{7} x ,$ and the line perpendicular to $f \left(x\right)$ that passes through the origin.

2.5

2.3 Exercise 16

Consider this scenario: A town’s population has been decreasing at a constant rate. In 2010 the population was 5,900. By 2012 the population had dropped to 4,700. Assume this trend continues.

Predict the population in 2016.

2.5

2.3 Exercise 18

Consider this scenario: A town’s population has been increased at a constant rate. In 2010 the population was 46,020. By 2012 the population had increased to 52,070. Assume this trend continues.

Predict the population in 2016.

2.5

2.3 Exercise 20

Consider this scenario: A town has an initial population of 75,000. It grows at a constant rate of 2,500 per year for 5 years.

Find the linear function that models the town’s population $P$ as a function of the year, $t ,$ where $t$ is the number of years since the model began.

2.5

2.3 Exercise 22

Consider this scenario: A town has an initial population of 75,000. It grows at a constant rate of 2,500 per year for 5 years.

If the function $P$ is graphed, find and interpret the x- and y-intercepts.

2.5

2.3 Exercise 24

Consider this scenario: A town has an initial population of 75,000. It grows at a constant rate of 2,500 per year for 5 years.

When will the output reached 100,000?

2.5

2.3 Exercise 26

Consider this scenario: The weight of a newborn is 7.5 pounds. The baby gained one-half pound a month for its first year.

Find the linear function that models the baby’s weight $W$ as a function of the age of the baby, in months, $t .$

2.5

2.3 Exercise 28

Consider this scenario: The weight of a newborn is 7.5 pounds. The baby gained one-half pound a month for its first year.

If the function $W$ is graphed, find and interpret the x- and y-intercepts.

2.5

2.3 Exercise 30

Consider this scenario: The weight of a newborn is 7.5 pounds. The baby gained one-half pound a month for its first year.

When did the baby weight 10.4 pounds?

2.5

2.3 Exercise 32

Consider this scenario: The number of people afflicted with the common cold in the winter months steadily decreased by 205 each year from 2005 until 2010. In 2005, 12,025 people were afflicted.

Find the linear function that models the number of people inflicted with the common cold $C$ as a function of the year, $t .$

2.5

2.3 Exercise 34

Consider this scenario: The number of people afflicted with the common cold in the winter months steadily decreased by 205 each year from 2005 until 2010. In 2005, 12,025 people were afflicted.

If the function $C$ is graphed, find and interpret the x- and y-intercepts.

2.5

2.3 Exercise 36

Consider this scenario: The number of people afflicted with the common cold in the winter months steadily decreased by 205 each year from 2005 until 2010. In 2005, 12,025 people were afflicted.

When will the number of people afflicted with the common cold reach 0?

2.5

2.3 Exercise 38

Find the linear function $y ,$ where $y$ depends on $t ,$ the number of years since 1980.

2.5

2.3 Exercise 40

Find and interpret the x-intercept.

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2.3 Exercise 42

Find the linear function $y ,$ where $y$ depends on $t ,$ the number of years since 1980.

2.5

2.3 Exercise 44

Find and interpret the x-intercept.

2.5

2.3 Exercise 46

In which state have home values increased at a higher rate?

2.5

2.3 Exercise 48

If we assume the linear trend existed before 1950 and continues after 2000, the two states’ median house values will be (or were) equal in what year? (The answer might be absurd.)

2.5

2.3 Exercise 50

If these trends were to continue, what would be the median home value in Indiana in 2010?

2.5

2.3 Exercise 52

In 2004, a school population was 1,001. By 2008 the population had grown to 1,697. Assume the population is changing linearly.

ⓐ How much did the population grow between the year 2004 and 2008?
ⓑ How long did it take the population to grow from 1,001 students to 1,697 students?
ⓓ What was the population in the year 2000?
ⓔ Find an equation for the p…

2.5

2.3 Exercise 54

A phone company has a monthly cellular plan where a customer pays a flat monthly fee and then a certain amount of money per minute used for voice and video calling. If a customer uses 410 minutes, the monthly cost will be $71.50. If the customer uses 720 minutes, the monthly cost will be $118.

ⓐ Find a linear equation for the monthly cost of the cell plan as a function of x, the number of monthly…

2.5

2.3 Exercise 56

In 1990, the moose population in a park was measured to be 4,360. By 1999, the population was measured again to be 5,880. Assume the population continues to change linearly.

ⓐ Find a formula for the moose population, P since 1991.
ⓑ What does your model predict the moose population to be in 2003?

2.5

2.3 Exercise 58

The Federal Helium Reserve held about 16 billion cubic feet of helium in 2010 and is being depleted by about 2.1 billion cubic feet each year.

ⓐ Give a linear equation for the remaining federal helium reserves, $R,$ in terms of $t,$ the number of years since 2010.
ⓑ In 2015, what will the helium reserves be?

2.5

2.3 Exercise 60

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2.3 Exercise 62

When hired at a new job selling jewelry, you are given two pay options:

Option A: Base salary of $17,000 a year with a commission of 12% of your sales
Option B: Base salary of $20,000 a year with a commission of 5% of your sales

How much jewelry would you need to sell for option A to produce a larger income?

2.5

2.3 Exercise 64

When hired at a new job selling electronics, you are given two pay options:

Option A: Base salary of $20,000 a year with a commission of 12% of your sales
Option B: Base salary of $26,000 a year with a commission of 3% of your sales

How much electronics would you need to sell for option A to produce a larger income?

2.5

Adapted from College Algebra © 2021 Rice University. Licensed under CC BY 4.0. Sources: College Algebra 2e, Algebra and Trigonometry 2e, Precalculus 2e, College Algebra with Corequisite Support 2e. Imported at revision 4922e46.