# optimization calculus

a real estate office manages 50 apartments in 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 be real estate office charge to maximize profits?

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1. let the number of \$25 increases be n
new rent = 900 + 25n
number rented = 50-n

profit = (50-n)(900+25n) - 75n
= 45000+275n - 25n^2

d(profit)/dn = 275 - 50n = 0 for max/min of profit
50n = 275
n = 275/50 = 5.5

The question is not clear whether the increases have to be in exact jumps of \$25.
if so, then n=5 or n=6 will yield the same profit
e.g.
if n=6, profit = 45000+1650-900 = 45750
if n=5 , profit = 45000+1375-625 = 45750

if a partial increase is allowed, then
n = 5.5
new rent = 1037.50
number rented = 50-5.5 = 44.5 ???? (makes no real sense)
but mathematically profit
= 45000+5.5(275)-25(5.5)^2 = 45756.25

I would go with either n=5 or n=6

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