Question

A real estate office handles 50 apartment units. When the rent is $720 per month, all units are occupied. However, on the average, for each $40 increase in rent, one unit becomes vacant. Each occupied unit requires an average of $48 per month for service and repairs. What rent should be charged to obtain a maximum profit?

Answer 1

If n is the number of apartments are occupied. Total Rent company is getting: $(720 + 40(50-n)) n - 48n$ 

We have to maximize the above equation.

$y=720n + 2000n - 40n^2 - 48n \\y=2672n-40n^2 \\y'=(2672n-40n^2)'=2672-80n \\y'=0 \\2672-80n=0 \\2672=80n \\n= 33.4 \approx 33$

$\text{Rent} = 720 + (50-33)40 = 720 + 680 = \ 1400$