Objectives
- Build linear models from verbal descriptions, and use the models to establish conclusions, including by contextualizing the meaning of slope and intercept parameters.
Section 3.5 Linear Models and Meanings (LF5)
Subsection 3.5.1 Activities
Remark 3.5.1.
We begin by revisiting Activity 2.2.14 from Section 2.2.
Activity 3.5.2.
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, \(A(w)\text{,}\) she has after \(w\) weeks can be modeled by the function
\begin{equation*}
A(w)=13+1.50w\text{.}
\end{equation*}
(a)
How much money will be in her piggy bank after 5 weeks?
Answer.
\(\$20.50\)
(b)
After how many weeks will she have \(\$40\) in her piggy bank?
Answer.
\(18\) weeks
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.Activity 3.5.4.
Jack bought a package containing 40 cookies. Each day he takes two in his lunch to work.
(a)
How many cookies are left in the package after 3 days?
(b)
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 |
|---|---|
| \(0\) | |
| \(2\) | |
| \(5\) | |
| \(10\) | |
| \(20\) |
(c)
What is the \(y\)-intercept? Explain what it represents in the context of the problem.
(d)
What is the rate of change? Explain what it represents in the context of the problem.
(e)
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!)
(f)
Find \(C(6)\text{.}\) Explain what that means in the context of the problem.
(g)
How many days will it take to empty the package? What does this correspond to on the graph?
Activity 3.5.5.
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 \(\$0.25\text{.}\)
(a)
In January, Daisy’s Doughnut Shop produced \(1000\) doughnuts. What was their total monthy cost to run the shop?
- \(\displaystyle \$2000.25\)
- \(\displaystyle \$2002.50\)
- \(\displaystyle \$2025.00\)
- \(\displaystyle \$2250.00\)
- \(\displaystyle \$4500.00\)
Answer.
D
(b)
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 |
|---|---|
| \(0\) | \(\$2000\) |
| \(500\) | \(\$2125\) |
| \(1000\) | \(\$2250\) |
| \(1500\) | \(\$2375\) |
| \(2000\) | \(\$2500\) |
(c)
What is the \(y\)-intercept? Explain what it represents in the context of the problem.
Answer.
\(\$2000\text{;}\) It is the fixed cost that must be paid regardless.
(d)
What is the rate of change? Explain what it represents in the context of the problem.
Answer.
\(0.25\text{;}\) This how much the total cost changes for each additional doughnut produced.
(e)
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.
\(C(x)=2000+0.25x\)
(f)
Find \(C(1300)\text{.}\) Explain what that means in the context of the problem.
Answer.
\(\$2325\text{;}\) The cost to produce \(1300\) doughnuts is \(\$2325\text{.}\)
(g)
Find the \(x\)-intercept. Explain what it means in the context of the problem.
Answer.
\(-8000\text{;}\) This doesn’t make sense in the context in the problem. Daisy can’t produce a negative number of doughnuts.
Activity 3.5.6.
A taxi costs \(\$5.00\) up front, and then charges \(\$0.73\) per mile traveled.
(a)
Write a linear function to model this situation.
Answer.
\(C(x)=5+0.73x\)
(b)
How much will it cost for a \(13\) mile taxi ride?
Answer.
\(\$14.49\)
(c)
If the taxi ride cost \(\$11.06\text{,}\) how many miles did it travel?
Answer.
\(8.3\) miles
Activity 3.5.7.
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.
(a)
Water freezes at \(0\)°\(C\) and \(32\)°\(F\text{.}\) Water boils at \(100\)°\(C\) and \(212\)°\(F\text{.}\) 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 \((100,212)\)
(b)
Use the two points to write a linear model for this situation. Use \(C\) and \(F\) as your variables.
Answer.
\(F=\frac{9}{5}C+32\)
(c)
If the temperature outside is \(25\)°\(C\text{,}\) what is the temperature in Fahrenheit?
Answer.
\(77\)°\(F\)
(d)
If the temperature outside is \(50\)°\(F\text{,}\) what is the temperature in Celsius?
Answer.
\(10\)°\(C\)
(e)
What temperature value is the same in Fahrenheit as it is in Celsius?
Answer.
\(-40\)°\(C=-40\)°\(F\)
Activity 3.5.8.
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.
(a)
Write a linear function \(S(x)\) that models the pricing structure for producing \(x\) shirts at Shirts-R-Us.
Answer.
\(S(x)=60+7x\)
(b)
Write a linear function \(G(x)\) that models the pricing structure for producing \(x\) shirts at Graphix!.
Answer.
\(G(x)=8.50x\)
(c)
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 \(G(1)=8.50\text{,}\) so less expensive at Graphix!.
Ten shirts: \(S(10) = 130\) and \(G(10)=85\text{,}\) so less expensive at Graphix!.
100 shirts: \(S(100) = 760\) and \(G(100)=850\text{,}\) so less expensive at Shirts-R-Us.
(d)
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.
Subsection 3.5.2 Videos
It would be great to include videos down here, like in the Calculus book!
