Probem58+Geneveive+Yutong+Paul

The manager of a 100-unit apartment complex knows from experience that all units will be occupied if the rent is $800 per month. A market survey suggests that, on average, one additional unit will remain vacant for each $10 increase in rent.What rent should the manager charge to maximize revenue?

Genevieve: Revenue is how much someone makes and is equal to rent x the # of occupied apartments ( r * x)

First, you have to build a function for the rent in terms of the number of occupied apartments. This is: r= 800 + 10(100-x) 800 is the initial rent when all the apartments are filled. 100-x is the number of unfilled apartments, and the 10 muliplied to that is the price increase.

So, you would see that if ALL apartments are sold (so x=100), r= 800.

Now, plug this into the revenue equation, which is R= r *x R= (800 + 10(100-x))x R= 1800x - 10x^2

Take the derivative, because the derivative will show us the maximum revenue, when setting it equal to zero.

dR/dx= 1800 - 20x set this equal to 0: 1800=20x, so x=90. This shows that the maximum revenue happens when there 90 apartments filled.

To find the exact rent when 90 apartments are filled, plug 90 into the rent equation. So r= 800 + 10(100-90) r= $900

So when rent is $900, 90 apartments will be filled, and the total revenue is??? Is this better than renting all of them at 800??