An apartment complex can rent all 200 of its one-bedroom apartments at a monthly rate of $600. for each $20 increase in rent, 4 additional apartments are left unoccupied. at what monthly rate should each apartment be rented in order to maximize the total rent collected?

The monthly rate should each apartment be rented is $ 800 in order to maximize the total rent collected

Solution:

Given that apartment complex can rent all 200 of its one-bedroom apartments at a monthly rate of $600

For rent:

Given that rent at monthly rate is $ 600

Also, For each $20 increase in rent, 4 additional apartments are left unoccupied

Therefore, for a given rent x,

Rent ⇒ 600 + 20x ( here plus sign denotes increase in rent)

Rooms ⇒ 200 - 4x ( here minus sign denotes 4 additional apartments are left unoccupied)

Now total revenue is given as:

R = (600 + 20x)(200 - 4x)

For maximum total rent to be collected,

Differentiate R with respect to x

To find the rate of each apartment be rented in order to maximize the total rent collected is:

Rent = 600 + 20x = 600 + 20(10) = 600 + 200 = $ 800

Thus monthly rate should each apartment be rented is $ 800