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.

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#
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Answer 2

The rate at which the cars should be rented to produce the maximum income is 42perday,andthemaximumincomeis42 per day, and the maximum income is 8,820 per day.

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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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