Rent-A-Reck Incorporated finds that it can rent 40 cars if it charges $\$ 60$ for a weekend. It estimates that for each $\$ 5$ price increase it will rent two fewer cars. What price should it charge to maximize its revenue?
\[
\$
\]

How many cars will it rent at this price?
$x$ cars
Show My Work (Optionan)

Step 1 ：The problem is asking for the price that will maximize the revenue. The revenue is the product of the price and the number of cars rented.

Step 2 ：We know that the number of cars rented decreases by 2 for every $5 increase in price. We can express the number of cars rented as a function of the price, and then express the revenue as a function of the price.

Step 3 ：Let's denote the price as \(p\) and the number of cars as \(c\). We can express \(c\) as a function of \(p\): \(c = 64 - 2\frac{p}{5}\).

Step 4 ：Then, the revenue \(r\) can be expressed as a function of \(p\): \(r = p(64 - 2\frac{p}{5})\).

Step 5 ：To find the maximum of this function, we can take the derivative of \(r\) with respect to \(p\) and set it equal to zero: \(r' = 64 - 4\frac{p}{5} = 0\).

Step 6 ：Solving this equation gives us the price that maximizes the revenue: \(p = 80\).

Step 7 ：Substituting \(p = 80\) into the equation for \(c\) gives us the number of cars that will be rented at this price: \(c = 64 - 2\frac{80}{5} = 32\).

Step 8 ：Final Answer: The price that should be charged to maximize its revenue is \(\boxed{\$80}\). The number of cars it will rent at this price is \(\boxed{32}\).