Calculus
posted by Deepa on .
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, all units require $75 in maintenance and repairs each month. How much rent should the real estate office charge to maximize profits?
I don't even know the answer to this one. I don't understand at all, please help me step by step

right now:
price of rent = 900
number of units rented = 50
Let the number of $25 increases be n
(e.g. If n= 2 , new rent is 900+2(25) = 950
if n = 5 , the new rent 900 + 5(25) = 1025
so the new rent = 900 + 25n
number rented = 50n
maintenance cost = 75(50n)
Profit = P = (900+25n)(50n)  75(50n)
= 45000 + 350n  25n^2  3750 + 75n
= 41250 + 425n  25n^2
d(profit)/dn = 425  50n
= 0 for a max of P
50n = 425
n = 8.5
The question did not say if increases are in whole multiples of 25 , but I will assume that. We could not rent 508.5 or 41.5 units.
when n = 8 or n = 9
if n = 8
number rented = 42
rent = 900+8(25) = 1100
maintenace cost = 75(42) = 3150
Profit = 42x1100  3150 = 43050
if n = 9
number rented = 41
rent = 1125
maintenance cost = 3075
Profit = 41x1125  3075 = 43050 , as expected. 
Wow thankyou sooooo much! you saved me xD