The monthly rate should each apartment be rented is $ 800 in order to maximize the total rent collected
<em><u>Solution:</u></em>
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)
![R = 120000 -2400x + 4000x - 80x^2](https://tex.z-dn.net/?f=R%20%3D%20120000%20-2400x%20%2B%204000x%20-%2080x%5E2)
For maximum total rent to be collected,
![\frac{dR}{dx} = 0](https://tex.z-dn.net/?f=%5Cfrac%7BdR%7D%7Bdx%7D%20%3D%200)
Differentiate R with respect to x
![0 - 2400 + 4000 - 160x = 0\\\\-160x = -1600\\\\x = 10](https://tex.z-dn.net/?f=0%20-%202400%20%2B%204000%20-%20160x%20%3D%200%5C%5C%5C%5C-160x%20%3D%20-1600%5C%5C%5C%5Cx%20%3D%2010)
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