Expert: Ahmed Salami - 11/9/2009

Question
Can you see if you can help me with the following problem?  Thank you.
7. (1 pt) The manager of a large apartment complex knows
from experience that 90 units will be occupied if the rent is 396
dollars per month. A market survey suggests that, on the average,
one additional unit will remain vacant for each 6 dollar
increase in rent. Similarly, one additional unit will be occupied
for each 6 dollar decrease in rent. What rent should the manager
charge to maximize revenue?______________


Answer
Hi Mikel,
Revenue is found by multiplying number of units N by the rent of each unit r.
R = N.r
If there are n increases of 6 dollars each in the rent, then
r = 396 + 6n
and
N = 90 - n
And so,
R = (90 - n)(396 + 6n)
 = 35640 + 144n - 6n²
With respect to n, the maximum revenue occurs when dR/dn = 0
But,
dR/dn =  144 - 12n
equating to zero,
144 - 12n = 0
12n = 144
n = 12
This means that for 12 increases each of 6 dollars in the rent, revenue is maximum.
The rent is then
r = 396 + 6(12)
 = 396 + 72
 = 468 dollars

Regards