maximizing revenue

[hephatsut1]

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 if the upper limit on the rent is $450 per month?

 

[tkhunny]

Isn't this a counting problem?

50*360 =
49*370 =
48*380 =
...

What do you get?

 

[soroban]

Hello, hephatsut1!

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.

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