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?

1 Answer
Jul 9, 2015

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#