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.
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?
(b)
After how many weeks will she have \(\$40\) in her piggy bank?
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)
(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 monthly cost to run the shop?
-
\(\displaystyle \$2000.25\)
-
\(\displaystyle \$2002.50\)
-
\(\displaystyle \$2025.00\)
-
\(\displaystyle \$2250.00\)
-
\(\displaystyle \$4500.00\)
(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.
(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 \(x\) represent the number of doughnuts produced and \(C(x)\) represent the total cost. (Hint: Use the previous two questions to help!)
(f)
Find \(C(1300)\text{.}\) Explain what that means in the context of the problem.
(g)
Find the \(x\)-intercept. Explain what it means in the context of the problem.
Activity 3.5.6.
(a)
Write a linear function to model this situation.
(b)
How much will it cost for a \(13\) mile taxi ride?
(c)
If the taxi ride cost \(\$11.06\text{,}\) how many miles did it travel?
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.
(b)
Use the two points to write a linear model for this situation. Use \(C\) and \(F\) as your variables.
(c)
(d)
(e)
What temperature value is the same in Fahrenheit as it is in Celsius?
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.
(b)
Write a linear function \(G(x)\) that models the pricing structure for producing \(x\) shirts at Graphix!.
(c)
At which business would it be less expensive to buy \(1\) shirt? \(10\) shirts? \(100\) shirts? Explain your reasoning.
(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.
Subsection 3.5.2 Exercises
Exercises available at
https://library.tbil.org/2025/precalculus/exercises/#/bank/LF5/
