3.2 Linear Functions Answer Key

Learning Objectives

In this department, you will:

• Identify steps for modeling and solving.
• Build linear models from verbal descriptions.
• Build systems of linear models.

Figure 1 (credit: EEK Photography/Flickr)

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

Identifying Steps to Model and Solve Problems

When modeling scenarios with linear functions and solving problems involving quantities with a constant rate of modify, nosotros typically follow the aforementioned problem strategies that we would use for any type of function. Let'southward briefly review them:

1. Identify changing quantities, and so define descriptive variables to correspond those quantities. When appropriate, sketch a movie or define a coordinate system.
2. Carefully read the problem to identify important information. Look for data that provides values for the variables or values for parts of the functional model, such every bit slope and initial value.
3. Carefully read the problem to determine what we are trying to detect, identify, solve, or interpret.
4. Place a solution pathway from the provided information to what we are trying to find. Oft this will involve checking and tracking units, building a table, or even finding a formula for the function beingness used to model the problem.
5. When needed, write a formula for the function.
6. Solve or evaluate the function using the formula.
7. Reverberate on whether your reply is reasonable for the given situation and whether it makes sense mathematically.
8. Conspicuously convey your effect using appropriate units, and respond in full sentences when necessary.

Building Linear Models

At present let'southward take a look at the educatee in Seattle. In her situation, there are ii changing quantities: time and coin. The amount of money she has remaining while on vacation depends on how long she stays. Nosotros can use this information to define our variables, including units.

• Output: $\mathrm{Yard},$ coin remaining, in dollars
• Input: $t,$ time, in weeks

So, the amount of money remaining depends on the number of weeks: $M(t)$

We can also identify the initial value and the rate of change.

• Initial Value: She saved $3,500, so $3,500 is the initial value for $G.$
• Rate of Change: She anticipates spending $400 each week, then –$400 per calendar week is the charge per unit of modify, or slope.

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 brand sense because she is spending coin each week.

The rate of change is abiding, and so nosotros tin first with the linear model $M\left(t\right)=mt+b.$ Then we can substitute the intercept and slope provided.

To find the $\mathrm{ten}\text{-}$ intercept, we prepare the output to zero, and solve for the input.

$$\begin{array}{l}0=-400t+3500\hfill \\ t=\frac{3500}{400}\hfill \\ =8.75\hfill \end{array}$$

The $x\text{-}$ intercept is 8.75 weeks. Because this represents the input value when the output volition be zero, we could say that Emily volition have no money left afterwards 8.75 weeks.

When modeling any real-life scenario with functions, there is typically a limited domain over which that model volition exist 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 cypher. A negative input value could refer to a number of weeks earlier she saved $three,500, just the scenario discussed poses the question one time she saved $3,500 considering this is when her trip and subsequent spending starts. It is too likely that this model is not valid after the $\mathrm{ten}\text{-}$ intercept, unless Emily volition use a credit card and goes into debt. The domain represents the set of input values, so the reasonable domain for this function is $0\le t\le \mathrm{8.75.}$

In the above example, we were given a written clarification of the state of affairs. We followed the steps of modeling a problem to analyze the information. However, the information provided may not e'er exist the aforementioned. Sometimes we might exist provided with an intercept. Other times we might be provided with an output value. We must be careful to analyze the information nosotros are given, and use it accordingly to build a linear model.

Using a Given Intercept to Build a Model

Some real-world problems provide the $y\text{-}$ intercept, which is the abiding or initial value. One time the $y\text{-}$ intercept is known, the $x\text{-}$ intercept tin can be calculated. Suppose, for example, that Hannah plans to pay off a no-interest loan from her parents. Her loan rest is $1,000. She plans to pay $250 per month until her residue is $0. The $y\text{-}$ intercept is the initial amount of her debt, or $1,000. The rate of change, or slope, is -$250 per month. Nosotros can so use the slope-intercept form and the given information to develop a linear model.

$$\begin{array}{l}f(x)=\mathrm{one\; thousand}x+b\hfill \\ =-250\mathrm{10}+1000\hfill \end{array}$$

Now we can ready the office equal to 0, and solve for $\mathrm{ten}$ to find the $\mathrm{10}\text{-}$ intercept.

$$\begin{array}{l}0=-250x+\mathrm{chiliad}\hfill \\ \mathrm{grand}=250x\hfill \\ \mathrm{four}=x\hfill \\ \mathrm{ten}=4\hfill \end{array}$$

The $x\text{-}$ intercept is the number of months it takes her to attain a balance of $0. The $x$ -intercept is 4 months, and so it will take Hannah four months to pay off her loan.

Using a Given Input and Output to Build a Model

Many real-world applications are not as direct as the ones we merely considered. Instead they require the states to identify some aspect of a linear role. Nosotros 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.

How To

Given a word problem that includes two pairs of input and output values, utilise the linear function to solve a trouble.

1. Place the input and output values.
2. Convert the data to 2 coordinate pairs.
3. Find the slope.
4. Write the linear model.
5. Use the model to make a prediction past evaluating the function at a given $x\text{-}$ value.
6. Use the model to place an $\mathrm{10}\text{-}$ value that results in a given $y\text{-}$ value.
7. Respond the question posed.

Case 1

Using a Linear Model to Investigate a Town'southward Population

A boondocks'southward population has been growing linearly. In 2004 the population was 6,200. By 2009 the population had grown to eight,100. Assume this tendency continues.

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

Attempt It #1

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

1. ⓐ Write a linear model to represent the toll $C$ of the company as a office of $x,$ the number of doughnuts produced.
2. ⓑ Find and translate the y-intercept.

Endeavour Information technology #2

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.

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

Using a Diagram to Model a Problem

It is useful for many existent-earth applications to describe 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 trouble is request for. Then, make up one's mind 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 pinnacle is helpful.

Example two

Using a Diagram to Model Altitude Walked

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

Q&A

Should I draw diagrams when given information based on a geometric shape?

Yes. Sketch the effigy and label the quantities and unknowns on the sketch.

Example 3

Using a Diagram to Model Distance betwixt Cities

There is a directly road leading from the town of Westborough to Agritown 30 miles eastward and 10 miles north. Partway down this road, it junctions with a second road, perpendicular to the beginning, leading to the town of Eastborough. If the boondocks of Eastborough is located xx miles directly eastward of the town of Westborough, how far is the road junction from Westborough?

Analysis

1 prissy use of linear models is to have advantage of the fact that the graphs of these functions are lines. This means real-world applications discussing maps need linear functions to model the distances between reference points.

Endeavor It #3

There is a direct 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?

Building Systems of Linear Models

Real-world situations including two or more than linear functions may be modeled with a system of linear equations. Call back, when solving a system of linear equations, nosotros are looking for points the 2 lines have in common. Typically, there are three types of answers possible, as shown in Figure 5.

Figure five

How To

Given a situation that represents a organisation of linear equations, write the system of equations and identify the solution.

1. Identify the input and output of each linear model.
2. Place the gradient and y-intercept of each linear model.
3. Observe the solution by setting the two linear functions equal to one another and solving for $\mathrm{ten},$ or notice the point of intersection on a graph.

Example 4

Edifice a System of Linear Models to Choose a Truck Rental Company

Jamal is choosing between 2 truck-rental companies. The first, Keep on Trucking, Inc., charges an up-front fee of $20, then 59 cents a mile. The second, Move It Your Style, charges an up-front end fee of $16, then 63 cents a mile⁴. When volition Continue on Trucking, Inc. exist the better choice for Jamal?

2.3 Department Exercises

Verbal

one.

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

ii .

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

3.

Explain how to translate the initial value in a discussion problem that uses a linear function.

4 .

Explain how to make up one's mind the slope in a word problem that uses a linear role.

Algebraic

5.

Discover the area of a parallelogram bounded by the y-axis, the line $\mathrm{ten}=3,$ the line $f(x)=\mathrm{i}+\mathrm{two}\mathrm{10},$ and the line parallel to $f(x)$ passing through $\left(\text{2},\text{ vii}\right).$

6 .

Observe the area of a triangle bounded by the x-axis, the line $f(x)=12–\frac{\mathrm{i}}{\mathrm{three}}\mathrm{10},$ and the line perpendicular to $f(x)$ that passes through the origin.

7.

Discover the area of a triangle bounded by the y-centrality, the line $f(x)=\mathrm{ix}–\frac{6}{7}\mathrm{ten},$ and the line perpendicular to $f(\mathrm{10})$ that passes through the origin.

8 .

Find the surface area of a parallelogram bounded by the x-centrality, the line $g(\mathrm{10})=\mathrm{two},$ the line $f(\mathrm{ten})=3\mathrm{10},$ and the line parallel to $f(x)$ passing through $(6,1).$

For the post-obit exercises, consider this scenario: A boondocks'southward 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.

ix.

Predict the population in 2016.

10 .

Place the twelvemonth in which the population volition accomplish 0.

For the post-obit 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.

11.

Predict the population in 2016.

12 .

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

For the following exercises, consider this scenario: A town has an initial population of 75,000. Information technology grows at a constant rate of 2,500 per yr for 5 years.

13.

Find the linear office that models the town's population $P$ equally a role of the year, $t,$ where $t$ is the number of years since the model began.

14 .

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

15.

If the function $P$ is graphed, detect and interpret the ten- and y-intercepts.

xvi .

If the function $P$ is graphed, detect and translate the slope of the function.

17.

When volition the output reached 100,000?

18 .

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 seven.5 pounds. The infant gained one-half pound a month for its get-go twelvemonth.

19.

Find the linear function that models the baby's weight $\mathrm{Westward}$ as a function of the age of the infant, in months, $t.$

20 .

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

21.

If the office $W$ is graphed, observe and interpret the 10- and y-intercepts.

22 .

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

23.

When did the baby weight 10.4 pounds?

24 .

What is the output when the input is half-dozen.ii? Translate your reply.

For the following exercises, consider this scenario: The number of people affected with the mutual cold in the wintertime months steadily decreased by 205 each year from 2005 until 2010. In 2005, 12,025 people were afflicted.

25.

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

26 .

Observe a reasonable domain and range for the part $C.$

27.

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

28 .

If the office $C$ is graphed, find and interpret the gradient of the function.

29.

When volition the output reach 0?

30 .

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

Graphical

For the following exercises, utilise the graph in Figure seven, which shows the turn a profit, $\text{y,}$ in thousands of dollars, of a company in a given year, $\text{t,}$ where $t$ represents the number of years since 1980.

Figure 7

31.

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

32 .

Notice and interpret the y-intercept.

33.

Find and interpret the 10-intercept.

34 .

Find and interpret the slope.

For the following exercises, use the graph in Effigy 8, which shows the profit, $y,$ in thousands of dollars, of a company in a given yr, $t,$ where $t$ represents the number of years since 1980.

Figure 8

35.

Observe the linear part $y,$ where $y$ depends on $t,$ the number of years since 1980.

36 .

Find and interpret the y-intercept.

37.

Discover and interpret the ten-intercept.

38 .

Observe and interpret the slope.

Numeric

For the following exercises, use the median dwelling house values in Mississippi and Hawaii (adjusted for inflation) shown in Table 2. Presume that the business firm values are changing linearly.

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

Table 2

39.

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

xl .

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

41.

If we assume the linear trend existed before 1950 and continues after 2000, the 2 states' median firm values volition be (or were) equal in what year? (The answer might be absurd.)

For the following exercises, use the median home values in Indiana and Alabama (adjusted for inflation) shown in Table 3. Assume that the house values are changing linearly.

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

Table 3

42 .

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

43.

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

44 .

If nosotros 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

45.

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

1. ⓐ How much did the population abound between the year 2004 and 2008?
2. ⓑ How long did it take the population to grow from 1,001 students to 1,697 students?
3. ⓒ What is the average population growth per year?
4. ⓓ What was the population in the year 2000?
5. ⓔ Detect an equation for the population, $P,$ of the school t years later on 2000.
6. ⓕ Using your equation, predict the population of the schoolhouse in 2011.

46 .

In 2003, a town'south population was ane,431. By 2007 the population had grown to two,134. Assume the population is changing linearly.

1. ⓐ How much did the population grow between the year 2003 and 2007?
2. ⓑ How long did information technology take the population to grow from 1,431 people to 2,134 people?
3. ⓒ What is the average population growth per year?
4. ⓓ What was the population in the year 2000?
5. ⓔ Find an equation for the population, $P$ of the town $t$ years subsequently 2000.
6. ⓕ Using your equation, predict the population of the boondocks in 2014.

47.

A telephone company has a monthly cellular programme where a customer pays a flat monthly fee and so a certain amount of money per minute used on the phone. If a customer uses 410 minutes, the monthly toll will be $71.l. If the customer uses 720 minutes, the monthly cost will be $118.

1. ⓐ Find a linear equation for the monthly cost of the cell plan as a function of x, the number of monthly minutes used.
2. ⓑ Interpret the slope and y-intercept of the equation.
3. ⓒ Use your equation to observe the total monthly price if 687 minutes are used.

48 .

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

1. ⓐ Find a linear equation for the monthly cost of the data plan every bit a part of $\mathrm{10}$, the number of MB used.
2. ⓑ Interpret the slope and y-intercept of the equation.
3. ⓒ Utilise your equation to find the total monthly cost if 250 MB are used.

49.

In 1991, the moose population in a park was measured to be 4,360. Past 1999, the population was measured again to be v,880. Assume the population continues to change linearly.

1. ⓐ Find a formula for the moose population, P since 1990.
2. ⓑ What does your model predict the moose population to exist in 2003?

50 .

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

1. ⓐ Find a formula for the owl population, $\mathrm{P.}$ Allow the input be years since 2003.
2. ⓑ What does your model predict the owl population to be in 2012?

51.

The Federal Helium Reserve held about 16 billion cubic feet of helium in 2010 and is existence depleted past most ii.one billion cubic feet each year.

1. ⓐ Requite a linear equation for the remaining federal helium reserves, $\mathrm{R,}$ in terms of $\mathrm{t,}$ the number of years since 2010.
2. ⓑ In 2015, what volition the helium reserves be?
3. ⓒ If the charge per unit of depletion doesn't change, in what year will the Federal Helium Reserve be depleted?

52 .

Suppose the earth's oil reserves in 2014 are 1,820 billion barrels. If, on average, the full reserves are decreasing past 25 billion barrels of oil each year:

1. ⓐ Give a linear equation for the remaining oil reserves, $\mathrm{R,}$ in terms of $\mathrm{t,}$ the number of years since now.
2. ⓑ Seven years from now, what volition the oil reserves exist?
3. ⓒ If the rate at which the reserves are decreasing is constant, when volition the earth's oil reserves be depleted?

53.

You are choosing between two different prepaid jail cell telephone plans. The first plan charges a rate of 26 cents per minute. The second plan charges a monthly fee of $xix.95 plus 11 cents per minute. How many minutes would you accept to use in a month in guild for the second plan to be preferable?

54 .

You are choosing between two different window washing companies. The showtime charges $5 per window. The 2nd charges a base fee of $xl plus $three per window. How many windows would you demand to have for the 2d company to be preferable?

55.

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

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

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

56 .

When hired at a new job selling electronics, yous are given 2 pay options:

• Option A: Base salary of $fourteen,000 a twelvemonth with a commission of ten% of your sales
• Pick B: Base of operations salary of $xix,000 a year with a committee of 4% of your sales

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

57.

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 committee of 12% of your sales
• Option B: Base of operations bacon of $26,000 a yr with a commission of 3% of your sales

How much electronics would yous demand to sell for option A to produce a larger income?

58 .

When hired at a new chore selling electronics, you are given ii pay options:

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

How much electronics would y'all need to sell for selection A to produce a larger income?

3.2 Linear Functions Answer Key,

Source: https://openstax.org/books/precalculus/pages/2-3-modeling-with-linear-functions

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