Respuesta :
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)
[tex]R = 120000 -2400x + 4000x - 80x^2[/tex]
For maximum total rent to be collected,
[tex]\frac{dR}{dx} = 0[/tex]
Differentiate R with respect to x
[tex]0 - 2400 + 4000 - 160x = 0\\\\-160x = -1600\\\\x = 10[/tex]
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
