A company handles an apartment building with 80 units. Experience has shown that if the rent for each of the units is $360 per month, all the 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 $600 per month?

Question

A company handles an apartment building with 80 units. Experience has shown that if the rent for each of the units is $360 per month, all the 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 $600 per month? ​

Quizplus · Accepted Answer

Let x be the number of $10 increases in the monthly rate.
Thus, the rent should be $(360+10x) per unit in order to fill (80-x) units.
Therefore, the total revenue from the building is:

Revenue = (360+10x)(80-x)

= -10x^2 + 200x + 28,800

Since the upper limit on the rent is $600 per month, we have:

360+10x ≤ 600

10x ≤ 240

x ≤ 24

Therefore, we only need to check the values of x from 0 to 24, and choose the value of x that gives the maximum revenue.

Revenue(0) = $28,800 (when all units are filled)

Revenue(1) = $28,890

Revenue(2) = $28,960

...

Revenue(24) = $24,000

The maximum revenue occurs when x = 8, which means that the rent should be $440 per unit.

Therefore, the best choice is C.

A Company Handles an Apartment Building with 80 Units