# How to write quadratic function given the following information?

## An apartment complex has 1600 units available of which 800 are currently rented for 300 dollars per month. A market survey indicated that a each five dollar decrease in monthly rent will result in 20 new renters. Write a function that models the monthly income, where x is the number of five dollar decreases in monthly rent. Then find the rent which yields the maximum monthly income, and what that income is.

Feb 20, 2018

$y = - 100 {x}^{2} - 4000 x + 246000$

#### Explanation:

Well first off, you have to figure out how to write all of the terms in terms of $x$. So here, there are two terms to figure out: the amount of money your current renters will pay you, plus the amount of rent your new renters will pay you. However, before we do this, we have to figure out a function for the rent, because it is going to change.

This part is pretty straightforward. The rent is going to start at $300 and continue to go down by $5 each time there is a decrease. So we get

$300 - 5 x$

as the rent with each decrease. This is because you start at $300 and take away $5 each time there is a decrease. This makes the next part fairly simple.

So now, we have to find out how many renters we have at a given time. So the function is going to be your current renters, plus the amount of new renters you will get. Since the original $800$ renters is always constant, it will not have a variable in our function.

However, you will get new $20$ new renters with each decrease, so this will be $20 x$ in the function. So our function for how many renters we have at any given time is

$800 + 20 x$

Now, finally, from logic, we know that the amount of renters at a given time $\left(800 + 20 x\right)$ times the rent at a given time $\left(300 - 5 x\right)$, because each person pays the rent, is equal to the total profit.

$y = \left(800 + 20 x\right) \left(300 - 5 x\right)$