The manager of a 100-unit apartment complex knows from experience that all units will be occupied if the rent is $400 per month. A market survey suggests that, on the average, one additional unit will remain vacant for each $5 increase in rent. What rent should the manager charge to maximize revenue?


this is my work so far:
So i want maximize the revenue so I need an expression for the revenue.

revenue = (number of units rented) (rent per unit)

100 * $400 = $40 000

let increments be x:

The number of units rented would then be 100 - x and the rent per unit would be $400 + $5x. Hence the revenue would be

(100 - x)($400 + $5x)

Then i took the derivative of that function above to maximize it and i got x=30.

then i plugged in 30 in the revenue function and i got 4900 dollars. i don't think that right because increasing the rent from 800 to 4900 is too much.

would equation would i substitute x=30 in then?

I'm sorry -- but I don't know how to figure this out algebraically -- but I made a chart showing the revenue for rents up to $450. With 90 apartments occupied, the monthly revenue is $40,500. At this point the revenue keeps increasing. However, at rents of $500 and 80 units occupied, the total revenue is back at $40,000.

as my total rent, i got 850 dollars.

the man will increase the rent by a maxium of 50 dollars.

i plugged in 30 into the revenue function and i got 4900.

4900/100 is 49 dollars per unit.

i rounded 49 to 50.

I think you mean that the highest rent would be $450 per unit (not $850).

I think there's something wrong with your numbers. However, you've found the right answer. The maximum rent should be $450 for 90 occupied apartments -- bringing in $40,500 a month to the apartment complex.

You have the right values at x=30 but you have to multiple 30 times 5 which is 150 added to the original rent making the maximum revenue coming when you have 30 vacant rooms and have 70 rooms occupied

Hope this helps

To find the rent that will maximize revenue, you correctly set up the revenue function as:

Revenue = (number of units rented) * (rent per unit)
Revenue = (100 - x) * ($400 + $5x)

To find the value of x that maximizes revenue, you need to find the critical points of the function. To do this, you should take the derivative of the revenue function with respect to x and set it equal to 0:

d(Revenue) / dx = (d/dx)(100 - x) * ($400 + $5x) + (100 - x) * (d/dx)($400 + $5x) = 0

Simplifying this equation will help us find the critical points.