TBIL-Precal Linear Models and Meanings (LF5)

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Section 3.5 Linear Models and Meanings (LF5)

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Objectives

Build linear models from verbal descriptions, and use the models to establish conclusions, including by contextualizing the meaning of slope and intercept parameters.

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Subsection 3.5.1 Activities

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Remark 3.5.1.

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We begin by revisiting Activity 2.2.14 from Section 2.2.

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Activity 3.5.2.

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Ellie has

$ 13

in her piggy bank, and she gets an additional

$ 1.50

each week for her allowance. Assuming she does not spend any money, the total amount of allowance,

A (w),

she has after

w

weeks can be modeled by the function

A (w) = 13 + 1.50 w .

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(a)

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How much money will be in her piggy bank after

5

weeks?

Answer.

$ 20.50

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(b)

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After how many weeks will she have

$ 40

in her piggy bank?

Answer.

18

weeks

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Remark 3.5.3.

The function in the previous activity is an example of a linear model. A linear model is a linear function that describes, or models, a real-life application. In this section we will build and use linear models.

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Activity 3.5.4.

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Jack bought a package containing

40

cookies. Each day he takes two in his lunch to work.

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(a)

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How many cookies are left in the package after

3

days?

$46$
$42$
$38$
$36$
$34$

Answer.

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(b)

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Fill out the following table to represent the number of cookies left in the package after the given number of days.

number of days	number of cookies left
$0$
$2$
$5$
$10$
$20$

Answer.

number of days	number of cookies left
$0$	$40$
$2$	$38$
$5$	$30$
$10$	$20$
$20$	$0$

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(c)

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What is the

y

-intercept? Explain what it represents in the context of the problem.

Answer.

40;

It represents the initial amount of cookies in the package.

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(d)

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What is the rate of change? Explain what it represents in the context of the problem.

Answer.

- 2;

It represents how the number of cookies in the package changes per day.

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(e)

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Write a linear function to model the situation. Let

d

represent the number of days elapsed and

C (d)

represent the number of cookies in the package. (Hint: Use the previous two questions to help!)

Answer.

C (d) = 40 - 2 d

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(f)

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Find

C (6) .

Explain what that means in the context of the problem.

Answer.

28;

After

6

days, there will be

28

cookies left in the package.

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(g)

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How many days will it take to empty the package? What does this correspond to on the graph?

Answer.

20;

This corresponds to the

x

-intercept.

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Activity 3.5.5.

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Daisy’s Doughnut Shop sells delicious doughnuts. Each month, they incur a fixed cost of

$ 2000

for rent, insurance, and other expenses. Then, for each doughnut they produce, it costs them an additional

$ 0.25 .

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(a)

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In January, Daisy’s Doughnut Shop produced

1000

doughnuts. What was their total monthy cost to run the shop?

$$ 2000.25$
$$ 2002.50$
$$ 2025.00$
$$ 2250.00$
$$ 4500.00$

Answer.

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(b)

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Fill out the following table to represent the cost for producing various amounts of doughnuts.

number of doughnuts	cost
$0$
$500$
$1000$
$1500$
$2000$

Answer.

number of doughnuts	cost
$0$	$$ 2000$
$500$	$$ 2125$
$1000$	$$ 2250$
$1500$	$$ 2375$
$2000$	$$ 2500$

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(c)

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What is the

y

-intercept? Explain what it represents in the context of the problem.

Answer.

$ 2000;

It is the fixed cost that must be paid regardless.

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(d)

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What is the rate of change? Explain what it represents in the context of the problem.

Answer.

0.25;

This how much the total cost changes for each additional doughnut produced.

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(e)

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Write a linear function to model the situation. Let

x

represent the number of doughnuts produced and

C (x)

represent the total cost. (Hint: Use the previous two questions to help!)

Answer.

C (x) = 2000 + 0.25 x

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(f)

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Find

C (1300) .

Explain what that means in the context of the problem.

Answer.

$ 2325;

The cost to produce

1300

doughnuts is

$ 2325 .

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(g)

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Find the

x

-intercept. Explain what it means in the context of the problem.

Answer.

- 8000;

This doesn’t make sense in the context in the problem. Daisy can’t produce a negative number of doughnuts.

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Activity 3.5.6.

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A taxi costs

$ 5.00

up front, and then charges

$ 0.73

per mile traveled.

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(a)

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Write a linear function to model this situation.

Answer.

C (x) = 5 + 0.73 x

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(b)

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How much will it cost for a

13

mile taxi ride?

Answer.

$ 14.49

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(c)

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If the taxi ride cost

$ 11.06,

how many miles did it travel?

Answer.

8.3

miles

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Activity 3.5.7.

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When reporting the weather, temperature is given in degrees Fahrenheit (

F

) or degrees Celsius (

C

). The two scales are related linearly, which means we can find a linear model to describe their relationship. This model lets us convert between the two scales.

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(a)

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Water freezes at

0

C

and

32

F .

Water boils at

100

C

and

212

F .

Use this information to write two ordered pairs.

Hint.

Choose Celsius to be your input value, and Fahrenheit to be the output value.

Answer.

(0, 32)

and

(100, 212)

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(b)

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Use the two points to write a linear model for this situation. Use

C

and

F

as your variables.

Answer.

F = \frac{9}{5} C + 32

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(c)

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If the temperature outside is

25

C,

what is the temperature in Fahrenheit?

Answer.

77

F

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(d)

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If the temperature outside is

50

F,

what is the temperature in Celsius?

Answer.

10

C

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(e)

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What temperature value is the same in Fahrenheit as it is in Celsius?

Answer.

- 40

C = - 40

F

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Activity 3.5.8.

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Erin needs to print t-shirts for her company retreat. She has found two businesses that produce quality shirts, but they have different pricing structures. Shirts-R-Us requires a

$ 60

set up fee, then charges

$ 7

for each shirt produced. Graphix! has no set up fee and charges

$ 8.50

per shirt produced.

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(a)

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Write a linear function

S (x)

that models the pricing structure for producing

x

shirts at Shirts-R-Us.

Answer.

S (x) = 60 + 7 x

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(b)

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Write a linear function

G (x)

that models the pricing structure for producing

x

shirts at Graphix!.

Answer.

G (x) = 8.50 x

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(c)

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At which business would it be less expensive to buy

1

shirt?

10

shirts?

100

shirts? Explain your reasoning.

Answer.

One shirt:

S (1) = 67

and

G (1) = 8.50,

so less expensive at Graphix!

10

shirts:

S (10) = 130

and

G (10) = 85,

so less expensive at Graphix!

100

shirts:

S (100) = 760

and

G (100) = 850,

so less expensive at Shirts-R-Us.

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(d)

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It depended on how many shirts were needed to determine which business was less expensive. For what range of number of shirts should Erin choose Shirts-R-Us and for what range should she choose Graphix!?

Hint.

Try finding where the cost is the same at both businesses. And looking at a graph may help as well!

Answer.

The businesses charge the same amount to produce

40

shirts. For less, choose Graphix! For more, choose Shirts-R-Us.

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Exercises 3.5.2 Exercises

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Exercises available at https://tbil.org/precalculus/preview/exercises/#/bank/LF5/.

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