A company handles an apartment building with 50 units. Experience has shown that if the rent...

A company handles an apartment building with 50 units. Experience has shown that if the rent for each of the units is $800 per month, all of the units will be filled, but 1 unit will become vacant for each $20 increase in this monthly rate. If the monthly cost of maintaining the apartment building is $12 per rented unit, what rent should be charged per month to maximize the profit?

Expert Solution

Let the number of units vacant be x

Number of buildings occupied = (50-x)

Monthly rate = (800+20x)

Cost of maintenance = 12(50-x)

Profit = Revenue - Cost = (50-x)(800+20x) - 12(50-x) = 40000 + 1000x - 800x - 20x^2 - 600 + 12x = -20x^2 + 212x + 39400

d/dx(Profit) = -40x + 212

Equating it to zero, we get x = 212/40 = 5.3

P(5) = -20(5)^2 + 212(5) + 39400 = 39960

P(6) = -20(6)^2 + 212(6)+ 39400 = 39952

Hence the maxima occurs for profit, when integer value is 5.

Monthly rent will be 800 + 5(20) = 900$

If decimal value is accepted, then try 800 + 5.3(20) = 906$

Note - Post any doubts/queries in comments section.

milcah answered 2 years ago

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