Calc HELP NEEDED!!!!?

A real estate office manages 50 apartments in a downtown building. When the rent is $900 per month, the units are occupied. For every $25 increase in rent, one unit becomes vacant. On average, each unit requires $75 in maintenance and repars each month. How much rent shour the res estate office charge to maximize profits?

The answer is $1100 or $1125(as in the txtbook). But i dont understand how to get taht answer.

So...
let's call the number of increases in price x
number of rooms: 50 - x
price per room: 900 + 25 x

profit would then be P(x) = (900 + 25x)(50 - x)
and cost is C(x) = 75(50 - x)

Total revenue is then R(x) = P(x) - C(x)
R(x) = (900 + 25x)(50 - x) - 75(50 - x)
R(x) = (50 - x)(900 + 25x - 75)
R(x) = (50 - x)(825 + 25x)
R(x) = 41250 + 1250x - 825x - 25x^2
R(x) = -25x^2 + 425x + 41250

now take the derivative to find when the max occurs:
R'(x) = -50x + 425
0 = -50x + 425
x = 8.5

In order to prove that this is the maximum value, the second derivative must be negative at x = 8.5, which it is:
R''(x) = -50
R''(8.5) = -50

since you can only have a whole number of increases in price, the maximum must occur at x = 8 or x = 9

Now, we need the price apartment at x = 8,9
price per room = 900 + 25x

900 + 25(8) = $1100
900 + 25(9) = $1125

If the rent is $900, your revenue and costs are

R=$900*50 = $45,000, C=$75*50 = $3,750

Profits are revenues - cost sp P = $45,000 - $3,750 = $41,250

There is an assumption here. Is the per unit cost only for occupied units or for all 50 regardless of if they are occupied.

Suppose you increase the rent by $25k. Then k units becomes vacant.

R = (900+25k)(50-k) = 45,000 + 1250k - 25k虏 - 900k
R = 25(1,800 + 14k - k虏)

You have 2 different cost functions

C1 = 75*50 = 3,750 (maintaining all units)
C2 = 75(50-k) = 3,750 - 75k (maintaining occupied units)

P1 = 25(1,800 + 14k - k虏) - 3,750
P2 = 25(1,800 + 14k - k虏) - 3,750 + 75k

Let's solve P1 for k first.

P1 = 25(1,650 + 14k - k虏)

Take the derivative of P1 with respect to k, set it to 0 and solve.

dP1/dk = 25(14) - 50k = 0

350 = 50k k--> 7

Since dP1/dx<0 we know this is a maximum.

This means we should increase the rent by $25*7=$175 to maximize profits under this assumption.

Now solve for P2

P2 = 25(1,650 + 17k - k虏)

dP2/dk = 25*17 - 50k = 0

425 = 50k

k=8.5

Since k is not an integer we should use either k=8 or k=9

P2(8) = 25(1,650 + 17(8) - (8)虏) = 43,050
P2(8.5) = 25(1,650 + 17(8.5) - (8.5)虏) = 43,056.25
P2(9) = 25(1,650 + 17(9) - (9)虏) = 43,050

Since P2(8)=P2(9) we can use either

k=8 implies a rent of $900+8*25 = $1,100
k=9 implies a rent of $900+9*25 = $1,125

You need to come up with an equation and then find the max

let r = rent
let u = units rented
let P = Profit

P = r * u - 75 * 50

u = 50 - (r -900)/25

P = 50 r - (r虏 - 900* r)/25 - 3750
= 86 r- r虏/25 -3750

dp/dr = 86 - 2r/25

to find a max or min the derivative = 0

0 = 86 - 2r/25
r = 86*25/2 = 1075

Either the book is wrong or I made a mistake that you should be able to find.

In addition to the 2 proposed answers consider that its not only the net money flow (gross revenue minus cost of maintenance) that is considered but also the percentage of net income over gross revenue or cost of maintenance.