Maximize profit, with a cost per unit included

[Ian McPherson]

I feel bad for posting right after I asked for help from another question, but I have to go to work and don't have two hours to sit tumbling this one around in my head the problem is


A company handles an apartment building with 50 units. Experience has shown that if the rent for each of the units is $720 per month, all of the units will be filled, but one unit will become vacant for each $20 increase in this monthly rate. If the monthly cost of maintaining the apartment building is $12 per rented unit, what rent should be charged per month to maximize the profit?

Is not : Profit = Revenue - Cost?
Revenue = (Number of apartments)(Amount made from filled apartments)?
Cost = (12x)?

So wouldn't the complete equation be:
P(x) = (50-x)(720+20x) - (12x)
The book says that the amount to increase it to should be $860, but when i solve I get x=20, raising it to $740. What am I doing wrong?

 

[galactus]

You appear to have the correct equation.

Upon differentiating, we get \(\displaystyle 268-40x\)

Setting this to 0 and solving for x gives x=6.7

Rounding up gives x=7. Which is correct and gives the 720+20(7)=860

 

[Ian McPherson]

Thank you for your time, Galactus, I see my error. I was setting the simplified P(x) function = 0 rather than the derivative. Also, I was plugging the number into the profit function rather than the small part for finding the rent. Thanks.