Question

A car rental agency has 150 cars. The owner finds that at a price of $48 per day, he can rent all the cars. For each $2 increase in price, the demand is less and 4 fewer cars are rented. For each car that is rented, there are routine maintenance costs of $5 per day. What rental charge will maximize profit?

Answer 1

Given that for each $2 increase in price, the demand is less and 4 fewer cars are rented.

Let x be the number of $2 increases in price, then the revenue from renting cars is given by

`(48 + 2x) * (150 - 4x)=7,200+108x-8x^2`.

Also, given that for each car that is rented, there are routine maintenance costs of $5 per day, then the total cost of renting cars is given by

`5(150-4x)=750-20x`

Profit is given by revenue - cost.
Thus, the profit from renting cars is given by

`(7,200+108x-8x^2)-(750-20x)=6,450+128x-8x^2`

For maximum profit, the differentiation of the profit function equals zero.
i.e.

`(d)/(dx) (6,450+128x-8x^2)=0 \\ \\ 128-16x=0 \\ \\ x= (128)/(16) =8`

The price of renting a car is given by 48 + 2x = 48 + 2(8) = 48 + 16 = 64.

Therefore, the rental charge will maximize profit is $64.