#### hephatsut1

##### New member

- Joined
- May 7, 2008

- Messages
- 1

- Thread starter hephatsut1
- Start date

- Joined
- May 7, 2008

- Messages
- 1

A company handles an apartment building with 50 units.

Experience has shown that if the rent for each of the units is $360 per month, all units will be filled,

but 1 unit will become vacant for each $10 increase in the monthly rate.

What rent should be charged to maximize the total revenue from the building?

Let \(\displaystyle x\) = number of $10 increases in monthly rent.

The month rent will be: .\(\displaystyle 360 + 10x\) dollars per unit.

Then \(\displaystyle x\) units will be vacant: .\(\displaystyle 50-x\) units are rented.

The revenue is: .\(\displaystyle R \;=\;(360 + 10x)(50-x)\) dollars.

We have:. \(\displaystyle R \;=\;18,000 + 140x - 10x^2\quad\hdots\) which we want to maximize.

Using Calculus, set \(\displaystyle R'\) equal to 0: .\(\displaystyle 140 - 20x \:=\:0\quad\Rightarrow\quad x \:=\:7\)

Hence, the rent should be raised seven times.

. . \(\displaystyle \text{The rent will be: }\;\$360 + 7(10) \;=\;\$430\text{ per month.}\)