optimization calculus
posted by Anonymous .
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?

let the number of $25 increases be n
new rent = 900 + 25n
number rented = 50n
profit = (50n)(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+1650900 = 45750
if n=5 , profit = 45000+1375625 = 45750
if a partial increase is allowed, then
n = 5.5
new rent = 1037.50
number rented = 505.5 = 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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