# Optimization - Maximizing Profit

I have been struggling with the problem below for quite some time now and no one can seem to figure it out, so I am asking it here. The question is as follows:

You own an apartment complex with 50 units, and if you charge 1000 dollars per month in rent, it is fully occupied For every 50 dollars that you increase the rent, one tenant moves out. It costs you 200 dollars per month to maintain an occupied apartment. (Empty apartments cost you nothing.) How much rent should you charge to maximize profit?

Any help is greatly appreciated!

## 2 Answers

Let $n$ = the number of times you increase the rent by 50 dollars. Let $r$ = the rent you are charging. Let $t$ = the number of tenants in the apartment. Let $p$ = total profit.

$$n=\frac{r-1000}{50}$$

$$t=-n+50$$

$$p=tr-200t$$

Optimize in terms of $r$ and you're done.

Let $T$ the number of tenants of the complex.

You own an apartment complex with $\color{red} {50}$ units, and if you charge 1000 dollars per month in rent, it is fully occupied For every $\color{blue} {50}$ dollars that you increase the rent, one tenant moves out. It costs you 200 dollars per month to maintain an occupied apartment. (Empty apartments cost you nothing.) How much rent should you charge to maximize profit?

The total income is $(1000 + \color{blue} {50}( \color{red} {50} - T) ) \times T$, and the cost of maintaining is $200 \times T$. Now optimize for $T$.