Question

All units of a 100 unit apartment complex are rented out when the monthly rent is set at$1000 per month. suppose that one unit becomes empty with each $10 increase in rent and that each occupied unit costs $200 per month in maintenance.what is the amount of rent per month that will maximize monthly profit?
a.) what is the equation we are trying to optimize? what is the domain?
b.)1.)what is the optimal number of apartments to rent?
2.)what is the optimal monthly rent per unit?
3.)what is the maximum profit?
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is the attached photo right if not please explain why
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Answer 1

Your picture has several issues.

1. x is not defined. It appears to be the number of apartments rented.
2. R(x) is defined two different ways. The first way, it looks like it is the revenue from a single apartment. The second way, it looks like it is the revenue from the entire apartment complex.
3. The derivative is in error. It should be -20x +2000. In any event, this is not the derivative you want. You're not trying to maximize revenue; you're trying to maximize profit.
4. It might be useful to write an equation for profit: P(x) = R(x) -200x = -10x² +1800x. Then when you go to maximize it, your derivative will be P'(x) = 0 = -20x +1800 ⇒ x = 90.

Your answer is correct, but the path you followed to get there has a few potholes.