please chech my answer optimization
posted by julie .
The manager of a 100unit 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
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