maximizing revenue

hephatsut1

New member
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 if the upper limit on the rent is $450 per month?
 
Isn't this a counting problem?

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

What do you get?
 
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 x\displaystyle x = number of $10 increases in monthly rent.

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

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

The revenue is: .R  =  (360+10x)(50x)\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 R\displaystyle R' equal to 0: .14020x=0x=7\displaystyle 140 - 20x \:=\:0\quad\Rightarrow\quad x \:=\:7


Hence, the rent should be raised seven times.

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

 
Top