Question 661·Medium·Math

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A video rental kiosk charges a fixed service fee plus a constant price per movie rented.

In one month, Maya rented 3 movies and paid $29.
The next month, she rented 5 movies and paid $39.

If the total cost $C$ (in dollars) is a linear function of the number of movies rented $m$ , which choice is the total cost when $m=8$ ?

A fixed fee plus a price per item makes a linear model: each extra item adds the same amount to the total. Subtract two bills to remove the unchanging fee, then divide by the change in item count to find the per-item price. Desmos can evaluate that rate and extend a known bill. Don't divide one whole bill by its item count; that bill includes the fee.

Hints

Hint 1
In a linear model, equal numbers of extra movies add equal amounts to the cost. The service fee is the same in both months, so what happens to it when you subtract the two bills?
Hint 2
A rate of change is the change in total cost divided by the change in movie count. Once you have the price per movie, start from a known bill and add the cost of only the extra movies.
Hint 3
The five-movie bill already includes the fixed service fee. To reach eight movies, count how many rentals are new and multiply by the per-movie rate. You don't need to charge the fee again.

Step-by-step

Approach 1: Extend a known bill
Step 1
Pair each movie count with its cost
The cost function $C$ pairs a movie count with its total bill, so keep each count with its bill: $C(3)=29$ and $C(5)=39$ . Here $C(3)$ means the total cost for $3$ movies, not the price of one movie.
Step 2
Find the price per extra movie
The rate of change is the extra dollars divided by the extra movies. Both bills include the same fee, so subtracting the bills removes the fee. Type $\frac{39-29}{5-3}$ in Desmos. It displays $5$ : the total rises by $5 for each additional movie. The $10 difference between the bills covers two movies, not one.
Step 3
Extend the five-movie bill to eight movies
Type $39+5(8-5)$ in Desmos. It displays $54$ . Starting from the five-movie total keeps the fee already in that bill; $8-5$ counts only the additional movies. The total cost for $8$ movies is $54. Choice C.

Approach 2: Build the full cost model
Step 1
Separate the two charges
Let $p$ be the price per movie and $f$ the fee. A linear model multiplies $p$ by the number of movies but charges $f$ once: $C(m)=pm+f$ .
Step 2
Write an equation for each bill
Put each movie count and total into the model. The three-movie bill gives $3p+f=29$ , and the five-movie bill gives $5p+f=39$ . Both equations use the same fixed fee $f$ .
Step 3
Find the movie price and fee
Type $[3p+f,5p+f]\sim[29,39]$ in Desmos. The matching positions tell it to fit both bills. Under PARAMETERS, the regression shows $p=5$ and $f=14$ : each movie costs $5, and the fee is $14.
Step 4
Evaluate the model for eight movies
Type $8p+f$ beneath the regression. Desmos displays $54$ . That is the cost of eight movies plus one fee, so the total is $54. Choice C.

Question 661·Medium·Math

Strategy

Hints

Step-by-step

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Strategy

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Step-by-step

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