What rent should the manager charge to maximize revenue?

[Neil6790]

The manager of a large apartment complex knows from experience that 90 units will be occupied if the rent is 400 dollars per month. A market survey suggests that, on the average, one additional unit will remain vacant for each 10 dollar increase in rent. Similarly, one additional unit will be occupied for each 10 dollar decrease in rent. What rent should the manager charge to maximize revenue?

All I know is that 90 units will give me $36000

What do I have to do after? I am so stuck on this problem I have no idea what to do.

 

[Mark44]

The manager of a large apartment complex knows from experience that 90 units will be occupied if the rent is 400 dollars per month. A market survey suggests that, on the average, one additional unit will remain vacant for each 10 dollar increase in rent. Similarly, one additional unit will be occupied for each 10 dollar decrease in rent. What rent should the manager charge to maximize revenue?

All I know is that 90 units will give me $36000

What do I have to do after? I am so stuck on this problem I have no idea what to do.

OK, so R(90) = 90* 400 = 36000.
How about R(89)? R(85)? R(91)? R(100)?

More generally, what is R(x), where x represents the number of units that are occupied?

In order to maximize the revenue function, you first need a revenue function.

 

[HallsofIvy]

The manager of a large apartment complex knows from experience that 90 units will be occupied if the rent is 400 dollars per month. A market survey suggests that, on the average, one additional unit will remain vacant for each 10 dollar increase in rent. Similarly, one additional unit will be occupied for each 10 dollar decrease in rent.

So the number of units, u, rented if the rent is r is 90- the number of 10 dollar increases in rent over 400. The increase is r- 400 and the number of 10 dollar increases in that is (r-400)/10. The number of units rented if the rent is r dollas per mont is u= 90- (r-400)/10

What rent should the manager charge to maximize revenue?

All I know is that 90 units will give me $36000

What do I have to do after? I am so stuck on this problem I have no idea what to do.

Find the number of units rented if the rent is r. That's what I gave above. Find the revenue in that case: multiply the number of units rented times r.

Find the maximum value of that revenue. You can do it by setting the derivative equal to 0 or, since this is a quadratic, by completing the square to find the vertex.