# Math

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A real estate company owns 218 efficiency apartments, which are fully occupied when the rent is \$940 /month. The company estimated that for each \$25 increase in rent, 5 apartments will become unoccupied. What rent should be charged so that the company will receive the maximum monthly income?

• Math -

let the number of \$25 increases be n

rent = 940 + 25n
number rented = 218-5n

income
= (940+25n)(218-5n)
= 204920 +750n - 125n^2

If you know Calculus:
d(income) = 750 - 250n = 0 for a max of income
250n=750
n = 3

if you don't know Calc
complete the square:
-125(n^2 - 6n + 9 - 9) + 204920
= -125( (n-3)^2 - 9) + 204920
= -125(n-3)^2 + 1125 + 204920
= -125(n-3)^2 + 206045
vertex is (3, 206045)

there should be 3 increases for a rent of 940+75 = 1015

or
the x of the vertex is -b/(2a) = -750/-250 = 3
for 3 increases.

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