How Much Should Rent Be Charged to Maximize Profit?

[polak333]

Homework Statement

A real estate office manages 50 apartments in a downtown building. When the rent is $900 per month, all the units are occupied. For ever $25 increase in rent, one unit becomes vacant. On average, all units require $75 in maintenance and repairs each month. How much rent should the real estate office charge to maximize profits.

2. The attempt at a solution

Total = (Number of apartments)(Monthly rent)
T = (50 + 25x)(900-75x)
T = (45000 - 3750x + 22500x - 1875x²)
T = -1875x² + 18750x + 45000
T' = -3750x + 18750
0 = -3750x + 18750
3750x = 18750
x = 5

I'm not sure if I'm actually doing this correctly or if I'm getting anywhere that's actually correct.

The answer is: $1100 or $1125.

Suggestions are welcome and helpful!
Thanks!

 

[Dick]

Your instincts are right to be "not sure" about that. It's pretty messed up. You should start by saying what "x" and "T" are. You want to maximize profits. Call that P. Now you need to express the numbers going into P in terms of the variables. The only variable I see here is the rent you charge.

 

[polak333]

Don't know if I understand.

 

[Dick]

Don't know if I understand.

Call the rent r. Profit "P" is money taken in minus money paid out. Money taken in is (number of apartment rented)*r. Can you write down an expression for the number of apartments rented in terms of r? The money paid out for maintenance should also be taken into account. Does that depend on the number of apartments rented. I'm not totally sure. How do you read the problem?

 

[cronxeh]

Version 1 assumes $75 in maintenance for every appartment, so 75*50 = $3750/month in fixed expenses monthly

Version 2 assumes $75 in maintenance for every rented appartment, so 75*(50-x) where x is number of $25 increases above $900 rent

 

[Dick]

cronxeh said that "x"=(number of $25 increases over the $900 rent). That's a good hint for polak333. Now what are the number of apartments rented in terms of x and the rent per apartment in terms of x? Your initial post isn't making much sense with that definition of x. If you had a different definition of x, you should say what it is.

 

[nailo1]

hey i just got the problem and i got a different answer, i went about it different but i think i got the right answer.

so i assumed $900 is the rent for a single apartment + 25*N for how many increases (0<N<50)

then (50-n) since every time N increases 50 goes down by 1

then i assume the average upkeep was for all 50 being 3750

i got the equation (900+25n)(50-n)-3750
first term =how much an apartment costs a month
second term = # of rooms
third term being upkeep

the f'(n)=25(50-n)+(-(900+25n))
f'(n)=1250-25n-900-25n
f'(n)=350-50n
-50n=-350
n=7

i got rent would be (900+25n)=1075

but that wasn't the answer in the back of the book so i checked it by "getting my hands dirty"
n=0,1,2,3,7,8,12
f(0)=41250
f(1)=41575
f(2)=41850
f(3)=42075
f(7)=42475
f(8)=42450
f(12)=41850

so according to my math 7 is the answer and i backed that up with a chart of answers. I've seen answers wrong in the back of the calculus book i got the problem, so i think it's the case here, and i also went over it several times in case the the equation was wrong.

but tell me if you find a problem

 

[Dick]

but tell me if you find a problem

Seems ok to me.

 

[nailo1]

Seems ok to me.

thanks it's good to get a second opinion.