Maximum Profit/Optimization problem (rent)

A

AGlas9837

Junior Member
Joined
Jan 23, 2008
Messages
57
A real estate office handles a 50-unit apartment complex. When the rent is $580/month, all units are occupied. For each $40 increase in rent, however, an average of 1 unit becomes vacant. Each occupied unit requires an average of $45 per month for service and repairs. What rent should be charged to obtain a maximum profit?

I used 50-49/580-620 to get a slope of -40 so that my profit equation is p-585=-40(x-50) or -40x+2585. If R = xp, then R=x(-40x+2585) or -40x^2+2585x. Cost = 45x so P=R-C becomes P=(-40x^2+2585x) - 45x or -40x^2+2540x.

dP/dx, then, is -80x+2540 giving x of 31.75. From here, I don't understand what to do with x which is a relative max. I can't plug x into any of my equations to get a plausible rent. What am I doing wrong? Thanks.
 
What does "x" stand for? (It can't be the units rented, since the slope is negative, making it most profitable to rent no units at all.)

How did you account for the change in price versus the change in number of units rented?

Please be complete. Thank you! :D

Eliz.
 
I didn't even think about that(!), so didn't associate the negative slope with a negative number of rentals, so now I'm back at square one because obviously everything I did was wrong. I was looking at the example of a similar problem where the point-slope equation was used and trying to use in this problem. Am I even going to be using the point-slope equation in this problem and, if so, what numbers should I be using in the equation?
 
AGlas9837 said:
I was looking at the example of a similar problem where the point-slope equation was used and trying to use in this problem.
Click to expand...
Instead of trying to force the exercise to fit a particular sort of equation, try using the translation skills you were taught back in algebra.

First, of course, you'll need to define your variable. What does "x" stand for?

Eliz.
 
I was using x as the number of apartments. I'm thinking now that it should be the amount of money charged per month or rent.
 
AGlas9837 said:
I was using x as the number of apartments. I'm thinking now that it should be the amount of money charged per month or rent.
Click to expand...
Re-read the exercise. What change is being measured? (Hint: What changes the amount charged and the number of rooms rented?)

If you're not sure, then start putting numbers into the problem, and seeing what you get:

Code: 
+----------+-------------+--------+-----------------------+
|increases | per-unit $  | units  | total income          |
+----------+-------------+--------+-----------------------+
|  n = 0   | 580         | 50     | (50)    (580)         |
|  n = 1   | 580 + 40    | 50 - 1 | (50 - 1)(580 + 40)    |
|  n = 2   | 580 + 2(40) | 50 - 2 | (50 - 2)(580 + 2(40)) |
|  n = 3   | 580 + 3(40) | 50 - 3 | (50 - 3)(580 + 3(40)) |
|  n = 4   | 580 + 4(40) | 50 - 4 |                       |
|  n = 5   | 580 + 5(40) |        |                       |
|  n = 6   |             |        |                       |
|          |             |        |                       |
|          |             |        |                       |
Continue until you see a pattern, and see where "x" could be most-usefully placed, just like you did back in algebra. The solution will be the vertex of a parabola, so either complete the square or use the formula to find the vertex. (Or, if you have studied calculus at all, you could try the derivative. But calculus is not needed for this exercise, so don't worry if you haven't done that yet.)

Eliz.
 
I would probably never have figured that out but my instructor gave us this function in class today. She said it was:
(580+40x)(50-x) - 45(50-x) so I multiplied out and got R=40x^2+1465x+26,750. I took the derivative, 80x+1465, set =0 and solved which gave me -18.31. This can't be right though. What have I done wrong this time?
 
AGlas9837 said:
I would probably never have figured that out but my instructor gave us this function in class today. She said it was:
(580+40x)(50-x) - 45(50-x) so I multiplied out and got R=40x^2+1465x+26,750. I took the derivative, 80x+1465, set =0 and solved which gave me -18.31. This can't be right though. What have I done wrong this time?
Click to expand...

Profit = (580 + 40x)(50 - x) - 45(50 - x) = -40x^2..... The x^2 term would be negative - so check and corret your expansion.
 
Actually, I had already caught that error, thanks. Still, I'm unsure if my answer, 18.31 is correct and, if so, where I "plug in" to get the maximum rent which can be charged. Am I just adding $18.31 to $580??
 
AGlas9837 said:
Actually, I had already caught that error, thanks. Still, I'm unsure if my answer, 18.31 is correct and, if so, where I "plug in" to get the maximum rent which can be charged. Am I just adding $18.31 to $580??
Click to expand...
You were unable to figure out the function, and then your instructor gave you the final version, so you would appear still not to understand what is going on, how that function was created. So please follow the instructions, provided in my earlier post.

Once you understand how the function is generated, what it means, you should be much closer to understanding what the "answer" gives you. :wink:

Eliz.
 
You have term in your equation - which is "(50-x)".

In there - what is the "unit" (dimension) of 50 - is it $? Is it # of people? Is it # of monkeys?

That will give you an idea about - what 'x' actually is.
 
The x in (580+40x) represents the number of increases in $40 increments in rent so it's related to money, but the x in (50-x) represents the decrease in the number of rentals for every $40 increase in rent so it's the number of rental units. So the equation doesn't make sense to me because in one part x seems to represent money and in the other, the number of rentals. But at this point, I can honestly say I give up and I don't care if I ever understand this problem! I really don't. I'm completely, utterly "over it."
 
AGlas9837 said:
The x in (580+40x) represents the number of increases in $40 increments in rent so it's related to money,

the term 40 x has following units:

40 - dollars/( rental unit)

x - rental unit

40x - dollars

x is the number of units that will be reduced - for each increase of rental of $40

but the x in (50-x) represents the decrease in the number of rentals for every $40 increase in rent so it's the number of rental units(correct - sort of - it is the number units that you'll lose for each increase of $40 rent/unit). So the equation doesn't make sense to me because in one part x seems to represent money and in the other, the number of rentals. But at this point, I can honestly say I give up and I don't care if I ever understand this problem! I really don't. I'm completely, utterly "over it."
Click to expand...
 
You must log in or register to reply here.
Top