# Calculus

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?

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1. 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 = 50-n
maintenance cost = 75(50-n)

Profit = P = (900+25n)(50-n) - 75(50-n)
= 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 50-8.5 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.

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