Modeling with Linear Functions

Emily is a college student who plans to spend a summer in Seattle. She has saved $3,500 for her trip and anticipates spending $400 each week on rent, food, and activities. How can we write a linear model to represent her situation? What would be the x-intercept, and what can she learn from it? To answer these and related questions, we can create a model using a linear function. Models such as this one can be extremely useful for analyzing relationships and making predictions based on those relationships. In this section, we will explore examples of linear function models.

Identifying Steps to Model and Solve Problems

When modeling scenarios with linear functions and solving problems involving quantities with a constant rate of change, we typically follow the same problem strategies that we would use for any type of function. Let’s briefly review them:

Building Linear Models

Now let’s take a look at the student in Seattle. In her situation, there are two changing quantities: time and money. The amount of money she has remaining while on vacation depends on how long she stays. We can use this information to define our variables, including units.

Notice that the unit of dollars per week matches the unit of our output variable divided by our input variable. Also, because the slope is negative, the linear function is decreasing. This should make sense because she is spending money each week.

The rate of change is constant, so we can start with the linear model

M (t) = m t + b .

intercept is 8.75 weeks. Because this represents the input value when the output will be zero, we could say that Emily will have no money left after 8.75 weeks.

When modeling any real-life scenario with functions, there is typically a limited domain over which that model will be valid—almost no trend continues indefinitely. Here the domain refers to the number of weeks. In this case, it doesn’t make sense to talk about input values less than zero. A negative input value could refer to a number of weeks before she saved $3,500, but the scenario discussed poses the question once she saved $3,500 because this is when her trip and subsequent spending starts. It is also likely that this model is not valid after the

x -

intercept, unless Emily will use a credit card and goes into debt. The domain represents the set of input values, so the reasonable domain for this function is

0 \leq t \leq 8.75.

In the above example, we were given a written description of the situation. We followed the steps of modeling a problem to analyze the information. However, the information provided may not always be the same. Sometimes we might be provided with an intercept. Other times we might be provided with an output value. We must be careful to analyze the information we are given, and use it appropriately to build a linear model.

Using a Given Intercept to Build a Model

intercept can be calculated. Suppose, for example, that Hannah plans to pay off a no-interest loan from her parents. Her loan balance is $1,000. She plans to pay $250 per month until her balance is $0. The

y -

intercept is the initial amount of her debt, or $1,000. The rate of change, or slope, is -$250 per month. We can then use the slope-intercept form and the given information to develop a linear model.

intercept is the number of months it takes her to reach a balance of $0. The

x -

Using a Given Input and Output to Build a Model

Many real-world applications are not as direct as the ones we just considered. Instead they require us to identify some aspect of a linear function. We might sometimes instead be asked to evaluate the linear model at a given input or set the equation of the linear model equal to a specified output.

Using a Diagram to Model a Problem

It is useful for many real-world applications to draw a picture to gain a sense of how the variables representing the input and output may be used to answer a question. To draw the picture, first consider what the problem is asking for. Then, determine the input and the output. The diagram should relate the variables. Often, geometrical shapes or figures are drawn. Distances are often traced out. If a right triangle is sketched, the Pythagorean Theorem relates the sides. If a rectangle is sketched, labeling width and height is helpful.

Using a Diagram to Model Distance Walked

Anna and Emanuel start at the same intersection. Anna walks east at 4 miles per hour while Emanuel walks south at 3 miles per hour. They are communicating with a two-way radio that has a range of 2 miles. How long after they start walking will they fall out of radio contact?

In essence, we can partially answer this question by saying they will fall out of radio contact when they are 2 miles apart, which leads us to ask a new question:

                                                         “How long will it take them to be 2 miles apart?”

In this problem, our changing quantities are time and position, but ultimately we need to know how long will it take for them to be 2 miles apart. We can see that time will be our input variable, so we’ll define our input and output variables.

Input: $t,$
time in hours.
Output: $A (t),$
distance in miles, and
$E (t),$
distance in miles

Because it is not obvious how to define our output variable, we’ll start by drawing a picture such as [link].

Initial Value: They both start at the same intersection so when $t = 0,$

the distance traveled by each person should also be 0. Thus the initial value for each is 0.

Rate of Change: Anna is walking 4 miles per hour and Emanuel is walking 3 miles per hour, which are both rates of change. The slope for $A$

is 4 and the slope for $E$

is 3.

Using those values, we can write formulas for the distance each person has walked.

\begin{array}{l} A (t) = 4 t \\ E (t) = 3 t \end{array}

For this problem, the distances from the starting point are important. To notate these, we can define a coordinate system, identifying the “starting point” at the intersection where they both started. Then we can use the variable, $A,$

which we introduced above, to represent Anna’s position, and define it to be a measurement from the starting point in the eastward direction. Likewise, can use the variable, $E,$

to represent Emanuel’s position, measured from the starting point in the southward direction. Note that in defining the coordinate system, we specified both the starting point of the measurement and the direction of measure.

We can then define a third variable, $D,$

to be the measurement of the distance between Anna and Emanuel. Showing the variables on the diagram is often helpful, as we can see from [link].

Recall that we need to know how long it takes for $D,$

the distance between them, to equal 2 miles. Notice that for any given input $t,$

the outputs $A (t), E (t),$

and $D (t)$

represent distances.

[link] shows us that we can use the Pythagorean Theorem because we have drawn a right angle.

Using the Pythagorean Theorem, we get:* * *

\begin{array}{l} D {(t)}^{2} = A {(t)}^{2} + E {(t)}^{2} \\ = {(4 t)}^{2} + {(3 t)}^{2} \\ = 16 t^{2} + 9 t^{2} \\ = 25 t^{2} \\ D (t) = \pm \sqrt{25 t^{2}} & Solve for D (t) using the square root \\ = \pm 5 \| t \| \end{array}

In this scenario we are considering only positive values of $t,$

so our distance $D (t)$

will always be positive. We can simplify this answer to $D (t) = 5 t .$

This means that the distance between Anna and Emanuel is also a linear function. Because $D$

is a linear function, we can now answer the question of when the distance between them will reach 2 miles. We will set the output $D (t) = 2$

and solve for $t .$

\begin{array}{l} D (t) = 2 \\ 5 t = 2 \\ t = \frac{2}{5} = 0.4 \end{array}

They will fall out of radio contact in 0.4 hours, or 24 minutes.

Building Systems of Linear Models

Real-world situations including two or more linear functions may be modeled with a system of linear equations. Remember, when solving a system of linear equations, we are looking for points the two lines have in common. Typically, there are three types of answers possible, as shown in [link].

Verbal

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

Determine the independent variable. This is the variable upon which the output depends.

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

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

To determine the initial value, find the output when the input is equal to zero.

Explain how to determine the slope in a word problem that uses a linear function.

Algebraic

Find the area of a parallelogram bounded by the y-axis, the line $x = 3,$

the line $f (x) = 1 + 2 x,$

and the line parallel to $f (x)$

passing through $(2, 7) .$

6 square units

Find the area of a triangle bounded by the x-axis, the line $f (x) = 12 - \frac{1}{3} x,$

and the line perpendicular to $f (x)$

that passes through the origin.

Find the area of a triangle bounded by the y-axis, the line $f (x) = 9 - \frac{6}{7} x,$

and the line perpendicular to $f (x)$

that passes through the origin.

20.012 square units

Find the area of a parallelogram bounded by the x-axis, the line $g (x) = 2,$

the line $f (x) = 3 x,$

and the line parallel to $f (x)$

passing through $(6, 1) .$

For the following exercises, 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 4,700. Assume this trend continues.

Predict the population in 2016.

2,300

Identify the year in which the population will reach 0.

For the following exercises, 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.

64,170

Identify the year in which the population will reach 75,000.

For the following exercises, 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.

P (t) = 75, 000 + 2,500 t

Find a reasonable domain and range for the function $P .$

If the function $P$

is graphed, find and interpret the x- and y-intercepts.

(–30, 0) Thirty years before the start of this model, the town had no citizens. (0, 75,000) Initially, the town had a population of 75,000.

If the function $P$

is graphed, find and interpret the slope of the function.

When will the output reached 100,000?

Ten years after the model began.

What is the output in the year 12 years from the onset of the model?

For the following exercises, 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, $t .$

W (t) = 0. 5 t + 7.5

Find a reasonable domain and range for the function $W$

If the function $W$

is graphed, find and interpret the x- and y-intercepts.

(- 15, 0)

: The x-intercept is not a plausible set of data for this model because it means the baby weighed 0 pounds 15 months prior to birth. $(0, 7.5)$: The baby weighed 7.5 pounds at birth.

If the function W is graphed, find and interpret the slope of the function.

When did the baby weight 10.4 pounds?

At age 5.8 months.

What is the output when the input is 6.2? Interpret your answer.

For the following exercises, 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, $t .$

C (t) = 12, 025 - 205 t

Find a reasonable domain and range for the function $C .$

If the function $C$

is graphed, find and interpret the x- and y-intercepts.

(58.7, 0)

: In roughly 59 years, the number of people inflicted with the common cold would be 0. $(0,12,025)$: Initially there were 12,025 people afflicted by the common cold.

If the function $C$

is graphed, find and interpret the slope of the function.

When will the output reach 0?

2064

In what year will the number of people be 9,700?

Graphical

For the following exercises, use the graph in [link], which shows the profit, $y,$

in thousands of dollars, of a company in a given year, $t,$

where $t$

represents the number of years since 1980.

Graph of a line from (15, 150) to (25, 130).

Find the linear function $y,$

where $y$

depends on $t,$

the number of years since 1980.

y = - 2 t +180

Find and interpret the y-intercept.

Find and interpret the x-intercept.

In 2070, the company’s profit will be zero.

Find and interpret the slope.

For the following exercises, use the graph in [link], which shows the profit, $y,$

in thousands of dollars, of a company in a given year, $t,$

where $t$

represents the number of years since 1980.

Graph of a line from (15, 150) to (25, 450).

Find the linear function $y,$

where $y$

depends on $t,$

the number of years since 1980.

y = 30 t - 300

Find and interpret the y-intercept.

Find and interpret the x-intercept.

(10, 0) In 1990, the profit earned zero profit.

Find and interpret the slope.

Numeric

For the following exercises, use the median home values in Mississippi and Hawaii (adjusted for inflation) shown in [link]. Assume that the house values are changing linearly.

Year	Mississippi	Hawaii
1950	$25,200	$74,400
2000	$71,400	$272,700

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

Hawaii

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

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.)

During the year 1933

For the following exercises, use the median home values in Indiana and Alabama (adjusted for inflation) shown in [link]. Assume that the house values are changing linearly.

Year	Indiana	Alabama
1950	$37,700	$27,100
2000	$94,300	$85,100

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

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

$105,620

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.)

Real-World Applications

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 is the average population growth per year?
What was the population in the year 2000?
Find an equation for the population, $P,$
of the school t years after 2000.
Using your equation, predict the population of the school in 2011.

696 people 4 years 174 people per year 305 people $P (t) = 305 + 174 t$ 2,219 people

In 2003, a town’s population was 1,431. By 2007 the population had grown to 2,134. Assume the population is changing linearly.

How much did the population grow between the year 2003 and 2007?
How long did it take the population to grow from 1,431 people to 2,134 people?
What is the average population growth per year?
What was the population in the year 2000?
Find an equation for the population, $P$
of the town
$t$
years after 2000.
Using your equation, predict the population of the town in 2014.

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 on the phone. 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 minutes used.
Interpret the slope and y-intercept of the equation.
Use your equation to find the total monthly cost if 687 minutes are used.

$C (x) = 0.15 x + 10$ The flat monthly fee is $10 and there is an additional $0.15 fee for each additional minute used $113.05

A phone company has a monthly cellular data plan where a customer pays a flat monthly fee of $10 and then a certain amount of money per megabyte (MB) of data used on the phone. If a customer uses 20 MB, the monthly cost will be $11.20. If the customer uses 130 MB, the monthly cost will be $17.80.

Find a linear equation for the monthly cost of the data plan as a function of $x$
, the number of MB used.
Interpret the slope and y-intercept of the equation.
Use your equation to find the total monthly cost if 250 MB are used.

In 1991, 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 1990.
What does your model predict the moose population to be in 2003?

$P (t) = 190 t + 4360$ 6,640 moose

In 2003, the owl population in a park was measured to be 340. By 2007, the population was measured again to be 285. The population changes linearly. Let the input be years since 1990.

Find a formula for the owl population, $P.$
Let the input be years since 2003.
What does your model predict the owl population to be in 2012?

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 depleted?

$R (t) = 16 - 2.1 t$ 5.5 billion cubic feet During the year 2017

Suppose the world’s oil reserves in 2014 are 1,820 billion barrels. If, on average, the total reserves are decreasing by 25 billion barrels of oil each year:

Give a linear equation for the remaining oil reserves, $R,$
in terms of
$t,$
the number of years since now.
Seven years from now, what will the oil reserves be?
If the rate at which the reserves are decreasing is constant, when will the world’s oil reserves be depleted?

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?

More than 133 minutes

You are choosing between two different window washing companies. The first charges $5 per window. The second charges a base fee of $40 plus $3 per window. How many windows would you need to have for the second company to be preferable?

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?

More than $42,857.14 worth of jewelry

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

Option A: Base salary of $14,000 a year with a commission of 10% of your sales
Option B: Base salary of $19,000 a year with a commission of 4% of your sales

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

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?

$66,666.67

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

Option A: Base salary of $10,000 a year with a commission of 9% of your sales
Option B: Base salary of $20,000 a year with a commission of 4% of your sales

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

Input	$d,$ distance driven in miles
Outputs	$K (d) :$ cost, in dollars, for renting from Keep on Trucking $M (d)$ cost, in dollars, for renting from Move It Your Way
Initial Value	Up-front fee: $K (0) = 20$ and $M (0) = 16$
Rate of Change	$K (d) = $ 0. 59$ /mile and $P (d) = $ 0. 63$ /mile

Modeling with Linear Functions

Identifying Steps to Model and Solve Problems

Building Linear Models

Using a Given Intercept to Build a Model

Using a Given Input and Output to Build a Model

Using a Diagram to Model a Problem

Building Systems of Linear Models

Verbal

Algebraic

Graphical

Numeric

Real-World Applications

Footnotes