A real estate office manages 50 apartments in a downtown building. When the rent is $900 per month, all the units are occupied. For every $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?

P(x) = R(x) - C(x)
This question I am stuck. I don't know how I would formulate the equations for revenue and cost. :|

Question

A real estate office manages 50 apartments in a downtown building. When the rent is $900 per month, all the units are occupied. For every $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?

P(x) = R(x) - C(x)
This question I am stuck. I don't know how I would formulate the equations for revenue and cost.  :|

amistre64 · Answer

50(75) are the costs....$3750

amistre64 · Answer

or is that only 75 per unit rented?

amistre64 · Answer

the rate of change is linear....

900,50
925,49..... for every 25 over we drop by 1

50=900/25+b

5=90/25+b

b = 5 - 90/25 = 7/5

rooms = cost/25 +(7/5) ??

amistre64 · Answer

900,50; and 925,49

50 + 900/25 = b lol   i forgot to negate the slope

amistre64 · Answer

50+36 = 86

y = -x/25+86

amistre64 · Answer

that is our demand function;.....

amistre64 · Answer

which can be rewritten as:

D(x) = (2150-x)/25

amistre64 · Answer

R * D(x) = total revenue  and that has to be at least equal to 3750 to break even right?

amistre64 · Answer

since R is the rent charged and x = rent in our demand function; lets do this..

x(2150-x)
---------  >= 3750
     25

amistre64 · Answer

2150x - x^2 >= 93750

0 >= x^2 -2150x + 93750 

+2150/2 is the x amount at the max :)

1075 = x

amistre64 · Answer

lets see if that amount works out....

amistre64 · Answer

2150-1075
---------- = rooms rented
      25

2150
1075
-----
1075/25 = 43

43(1075)>= 3750

1045
    43
-----
 4180
   3135
-------
 $44935 >= 3750...well, we make a profit at least :)

amistre64 · Answer

44935
  3750
------
 41185 in profits

amistre64 · Answer

we can trey that with 25 higher and 25 lower to see if we are right lol

amistre64 · Answer

2150-1100 
---------- = rooms rented 
       25 


2150
1100
-----
1050/25 = 42 ...really?  i had to figure it out that way lol...im an idiot :)

of course 25 higher is one less lol

amistre64 · Answer

1100
    42
-----
4400
  2200
-------
46200
......................... see?
1050
    44
-----
4200
  4200
------
46200

amistre64 · Answer

DOH!!...now wheres my mistake lol

amistre64 · Answer

1075
    43
-----
4300
  3225
------
46225 ...thats better lol, i forgot how to multiply