# 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 ##### Moderator Staff member Isn't this a counting problem? 50*360 = 49*370 = 48*380 = ... What do you get? #### soroban ##### Elite Member 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.

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