The manager of a large apartment complex knows from experience that 100 units will be occupied if the rent is 416 dollars per month. A market survey suggests that, on the average, one additional unit will remain vacant for each 8 dollar increase in rent. Similarly, one additional unit will be occupied for each 8 dollar decrease in rent. What rent should the manager charge to maximize revenue?

Question

The manager of a large apartment complex knows from experience that 100 units will be occupied if the rent is 416 dollars per month. A market survey suggests that, on the average, one additional unit will remain vacant for each 8 dollar increase in rent. Similarly, one additional unit will be occupied for each 8 dollar decrease in rent. What rent should the manager charge to maximize revenue?

I was given the answer 46208 but that answer is incorrect. Can someone help me? Please

Answer 1

let the number of $8 increments be n

so number occupied = 100-n
cost per unit = 416+8n

Revenue = R = (100-n)(416+8n)
= 41600+ 800n - 416n - 8n^
= 41600 + 384n - 8n^2

dR/dn = 384 - 16n = 0 for a max of R
16n = 384
n = 24

so the rent should be 416+8(24) = $608
with 100 - 24 or 76 units rented for a max
revenue of 76(608) or $46208

The answer is correct for the max profit, but it asked for the rental cost to obtain that , which
was $608 per unit.
for a max revenue of
41600 + 384(24) - 8(24)^2 or $