I need quick help with this math problem................?

Question

A car rental agency rents 200 cars per day at a rate of $30 per day. For each $1 increase in rate, 5 fewer cars are rented. At what rate should the cars be rented to produce the maximum income? What is the maximum income?


This is what I did:

Price (p) = 30-1n
# (x) = 200-5n

Revenue R(n) = (x)(p)

R(n) = (200-5n)(30-1n) = 5n^2 - 350n + 6,000
R'(n) = 10n - 350

10n - 350 = 0
n = 35

Cars should be rented at a rate of 35 to produce max income.

To get the maximum income, I tried plugging 35 into the revenue formula:

R(35) = 5(35)^2 - 350(35) + 6,000

I ended up getting -125. That doesn't make sense.
The answer is supposed to be $6,125.

What am I doing wrong??

Answer 1

I just love how you show your steps. This is a teacher's dream!!!

Your mistake is in the price.

Instead of P = 30 - n, it should be P = 30 + n.

Remember, as the prices INCREASES by $1, the number rented DECREASES by 5.

R(n) = -5n² + 50n + 6000
R'(n) = -10n + 50

n = 5

R(5) = $6,125





In your original solution, n = 35 would mean the company would give each customer $5 for renting.. Hardly a way to make a profit!!! They would also rent 25 cars. Therefore, there revenue would be -$125 as you came up with.


In the actual solution... the company should charge $35. They would then rent out 175 cars, making a maximum profit of $6,125.