Optimization

An apartment complex has 60 unites. Currently the price is $640 per month and all units are occupied. For each $15 increase in price, one unit becomes vacant. Each occupied unit costs the apartment company $40 per month for service. How much should they charge in order to maximize profit?

I am really just stuck on creating the equations, the actual algebra shouldn't be too much of an issue.

Question

Optimization

An apartment complex has 60 unites. Currently the price is $640 per month and all units are occupied. For each $15 increase in price, one unit becomes vacant. Each occupied unit costs the apartment company $40 per month for service. How much should they charge in order to maximize profit?

I am really just stuck on creating the equations, the actual algebra shouldn't be too much of an issue.

anonymous · Answer

The number of occupied apartments generating cash, multiplied by the profit per apartment is equal to the gross profit per month.

The following expression, consisting of two factors, computes the gross profit for a month where n is the number $15 rent increases added to the base rental rate of $640/month.  $40 is a fixed service overhead cost per occupied apartment.

profit = ( 60 - n)( 640 + 15*n -40 ) = $$-15 n^2+300 n+36000 $$When n is zero, 60 apartments generate $36000 profit per month.

A plot of the profit function is attached.  A casual glance at the curve indicates the the maximum profit occurs at around 10 rate increases.

Application of the calculus will confirm the maximum profit occurs at 10.