Maximize revenue

[legacyofpiracy]

I'm stuck yet again on these optimization problems, hopefully sooner or later I will get a hang of them. Untill that time could anyone nudge me along on this problem? Any help would be greatly appreciated.


The manager of a large apartment complex knows from experience that 90 units will be occupied if the rent is 350 dollars per month. A market survey suggests that, on the average, one additional unit will remain vacant for each 7 dollar increase in rent. Similarly, one additional unit will be occupied for each 7 dollar decrease in rent. What rent should the manager charge to maximize revenue?

 

[soroban]

Hello, legacyofpiracy!

This is a classic max/min problem, but unlike any others you've seen.
Some baby-talk may be in order . . .

The manager of an apartment complex knows from experience that 90 units will be occupied if the rent is $350/month.
A market survey suggests that, on the average, one additional unit will remain vacant for each $7 increase in rent.
Similarly, one additional unit will be occupied for each $7 decrease in rent.
What rent should the manager charge to maximize revenue?

Right now, he charges $350/month and 90 units are rented.
His total revenue is: \(\displaystyle \$350\,\times\,90\:=\: \$31,500\)
. . Can he do any better?

Let \(\displaystyle x\) = number of $7 increases in monthly rent.
The new monthly rent will be: \(\displaystyle 350\,+\,7x\) dollars.
He will lose \(\displaystyle x\) tenants; he will have: \(\displaystyle 90\,-\,x\) units rented.

His revenue will be: .\(\displaystyle R\:=\350\,+\,7x)(90\,-\,x)\) dollars.

And that is the function we want to maximize.

 

[legacyofpiracy]

Thank you so much Soroban you have been so helpful, I really appreciate it!