# Thread: Pre Calculus - building manager

1. ## Pre Calculus - building manager

The question:

You are the manager of a 100-unit apartment building. The average monthly maintenance cost per occupied unit is $100. You ordered a survey that revealed that all units can be rented when the rent, r is$900, and that one unit will become vacant with each $10 increase in rent. a) Find a fuction that gives you the number of units rented n, as a function of r. b) The revenue is the total amount of money you receive by renting the apartments. Find the revenue fuction. c) The net cash intake is the revenue minus the maintenance. Find a formula for the net cash intake. d) Determine the rent that will maximize the net cash intake. The problem is I don't know where to start. There are no examples of this particular problem in my book (that I nor my classmates can find) and the first part seems too simple. Could the function for a be as simple as: a) f(r) = rn; or perhaps f(r) = r100; or f(r) = r100 - 1000 b) R(n) = 900n -1000 c)C(n) = (910)(n-1) - 1000-1 R=Revenue, C=Cash I don't know the best way to approach this problem. 2. Originally Posted by allanobrien334 The question: You are the manager of a 100-unit apartment building. The average monthly maintenance cost per occupied unit is$100. You ordered a survey that revealed that all units can be rented when the rent, r is $900, and that one unit will become vacant with each$10 increase in rent.

a) Find a function that gives you the number of units rented n, as a function of r.

n = 100 - (r-900)/10 , r > 900

b) The revenue is the total amount of money you receive by renting the apartments. Find the revenue function.

R = rn = r[100 - (r-900)/10]

c) The net cash intake is the revenue minus the maintenance. Find a formula for the net cash intake.

C = rn - 100n = (r - 100)n = (r - 100)[100 - (r-900)/10]

d) Determine the rent that will maximize the net cash intake.

the function C is quadratic with a negative leading coefficient ... max will be at the vertex.
ok?

3. So far I have:
a) rent function = 900
+ 10x
b) the unit function = 100 x
c)Revenue function = (900 + 10x)(100 x)
thus R(x) = −10x2 + 100x + 90,000.

-b/2a = 5

so R(5) = 90,250 max revenue - maintanence

95 units * 100 = 9500
so 90250 -9500 = 80750 total net cash

4. Originally Posted by allanobrien334
The question:

You are the manager of a 100-unit apartment building. The average monthly maintenance cost per occupied unit is $100. You ordered a survey that revealed that all units can be rented when the rent, r is$900, and that one unit will become vacant with each $10 increase in rent. a) Find a fuction that gives you the number of units rented n, as a function of r. b) The revenue is the total amount of money you receive by renting the apartments. Find the revenue fuction. c) The net cash intake is the revenue minus the maintenance. Find a formula for the net cash intake. d) Determine the rent that will maximize the net cash intake. The problem is I don't know where to start. There are no examples of this particular problem in my book (that I nor my classmates can find) and the first part seems too simple. Could the function for a be as simple as: a) f(r) = rn; or perhaps f(r) = r100; or f(r) = r100 - 1000 Do any of those satify the conditions: f(900)= 100 and f(910)= 99? Since your function is supposed to give n= f(r), it cannot be (a). If f(r)= r(100), f(900)= 900(100)= 90000, not 100 so it is not (b) and if f(r)= r100- 1000= (900)(100)- 1000= 89000, so it is not c. Hint f(r) is a linear function and can be written as f(r)= Ar+ B for some numbers A and B. Use F(900)= 900A+ B= 100, F(910)= 910A+ B= 99 to find A and B. b) R(n) = 900n -1000 The revenue function is the number of apartments rented times the rent for each apartment. R(n,r)= nr. You can use the function n= f(r) you got in (a) to find r as a function of n and put into the function to find R(n) as a function of n only. But it is simpler, and better, to just replace n with f(r) from (a) so you have R(r) as a function of r. [quote]c)C(n) = (910)(n-1) - 1000-1 This is, of course, just R minus the maintenance cost. You are told that the maintenance cost of each rented appartment is$100 (presumably there is no cost of maintaining an unrented apartment) so the cost is 100r.
Subtract that from R(r) from (b). (Again, it is better to have C as a function of r, not n.)

R=Revenue, C=Cash
I don't know the best way to approach this problem.
You don't mention "(d) Determine the rent that will maximize the net cash intake."
This is the reason I recommend having first R and then C in terms of r, not n. You want to find the value of r that makes C as large as possible. You should have gotten a quadratic for C(r) with the coefficient of $r^2$ negative. Its graph is a parabola opening downward so the vertex is its highest value. Find that highest value by completing the square.