Question

A car rental company can rent 60 cars if it charges $80 for a weekend. They estimate that for each $10 price increase, they will sell five fewer cars. What price should they charge to maximize their revenue? How many cars will they rent at that price? (Let x = the number of $10 increases)

a) $100 per weekend, 40 cars rented
b) $70 per weekend, 45 cars rented
c) $90 per weekend, 35 cars rented
d) $110 per weekend, 30 cars rented

Answer 1

To maximize revenue, the car rental company should charge $90 per weekend and rent out 35 cars. This option will result in the highest possible revenue.

Step-by-step explanation:

To maximize revenue, the car rental company needs to find the price that will result in the highest number of cars rented. From the given information, we know that for each $10 price increase, there will be five fewer cars rented. Let's use x to represent the number of $10 price increases.

So, if the original price is $80, the price after x increases will be $80 + $10x. The number of cars rented will be 60 - 5x. Therefore, the revenue can be calculated by multiplying the price by the number of cars rented: revenue = ($80 + $10x) * (60 - 5x).

To find the price that maximizes revenue, we can take the derivative of the revenue function and set it equal to zero. By solving this equation, we can find the value of x which represents the number of $10 price increases. Substituting this value of x back into the revenue function will give us the maximum revenue.

Using this method, we can calculate the revenue for each option:

1. Option a) $100 per weekend, 40 cars rented: revenue = ($100 + $10 * 2) * (60 - 5 * 2) = $6000
2. Option b) $70 per weekend, 45 cars rented: revenue = ($70 + $10 * 1) * (60 - 5 * 1) = $4100
3. Option c) $90 per weekend, 35 cars rented: revenue = ($90 + $10 * 3) * (60 - 5 * 3) = $6000
4. Option d) $110 per weekend, 30 cars rented: revenue = ($110 + $10 * 4) * (60 - 5 * 4) = $5600

Therefore, option c) $90 per weekend with 35 cars rented will maximize the revenue.