Question

A car rental agency rents 220 cars per day at a rate of 27 dollars per day. For each 1 dollar increase in the daily rate, 5 fewer cars are rented. At what rate should the cars be rented to produce the maximum income, and what is the maximum income?

Answer 1

Assuming an even dollar rental is required;
The cars should be rented at $36 per day for a maximum income of $6300 per day.

Explanation:

If the daily rental is increased by $`x`
then
Rental: `R(x) =(27+x)` dollars per car-day
Number of cars rented: `N(x) =(220-5x)`
and
Income: `I(x) =(27+x)(220-5x) = 5840+85x-5x^2` dollars/day

The maximum will be achieved when the derivative of `I(x)` is zero.

`(d I(x))/(dx) = 85-10x = 0`

`rArr x = 8.5`

For an even dollar rental amount, and increase of $8/day or $9/day will generate the same income.
So `$27+$8 = $35`/day
or
`$27+$9 = $36`/day
would both be valid answers.
However, $36/day involves renting fewer cars and thus reduced expenses.

Using basic substitution and arithmetic
`color(white)("XXXX")` `I(9) = 6300`