Can somebody tell me what I did wrong in this financial mathematics problem?

A Real estate office manages 50 apartments in a downtown building. When the rent is $900 per month, all the units are occupied. For every $25 increase in rent, one unit becomes vacant. On average, each unit requires $75 in maintenance and repairs each month. How much rent should the real estate office charge to maximize profits?

I determined the cost function to be:
C(x) = 75(50) = 3750 (cause you have to pay for maintenance no matter what)

I determined the revenue function to be:
R(x) = (50-x)(900+25x)

When maximizing profit (P(x) = R(x) - C(x)) using a derivative, I get an x value of 7, meaning the rent should be $1075 to max out profit.

My answer sheet says otherwise: $1100 or $1125

Apparently, according to an answerer before, my cost function is wrong. I'm only asking this question again because I really need to fix it quick! Can someone PLEASE tell me what I did wrong?

My Answer: $ 1075
Sheet Answer: $ 1100 or $1125

Thanks James! I at least know that unused apartments aren't payed for in terms of maintenance. That still doesn't help me in figuring out this freakin cost function though. All I know is that it must be a third degree polynomial because that is the only way that the answer could possibly have two answers.

75(50 - x) is the cost function right?

Here's what I understand...

If Revenue (money you will make) is given by
(50 - x)(900 + 25x)
and Cost is given by
(75)(50 - x)


C: 3750 - 75x
R: 45000 + 1250x - 900x - 25x^2

-25x^2 - 900x + 1250x + 45000 - (3750 - 75x)
-25x^2 + 425x + 41250

P'(x) = -50x + 425 = 0
x = 8.5

you can interpret 8.5 as being 8 or 9

I just figured it out in Excel. (I know, it's a bit like cheating because it's not mathematically derived; I'll leave that to you.) To get the sheet answer of $1100 or $1125, they are assuming that you do NOT have to pay maintenance on unrented apartments. So your maintenance cost decreases as you increase rent, since you're losing one unit for every $25 increase.

And you're right, if maintenance costs are stable at $3750, then your optimal rent is $1075, giving you 43 occupied units.

James