# How Much Should Rent Be Charged to Maximize Profit?

• polak333
In summary, a real estate office manages 50 apartments in a downtown building with a rent of $900 per month. For every$25 increase in rent, one unit becomes vacant. On average, all units require $75 in maintenance and repairs each month. To maximize profits, the real estate office should charge$1100 or $1125 in rent. 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.

Thanks!

Last edited:
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.

Don't know if I understand.

polak333 said:
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?

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 Last edited: 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. Last edited: 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

nailo1 said:
but tell me if you find a problem

Seems ok to me.

Dick said:
Seems ok to me.

thanks it's good to get a second opinion.

## What is the "Optimization Rent Problem"?

The Optimization Rent Problem refers to the mathematical optimization problem of finding the most efficient way to allocate resources, such as land or space, in order to maximize profits or benefits.

## What are the key factors to consider when solving the Optimization Rent Problem?

The key factors to consider when solving the Optimization Rent Problem include the available resources, the desired outcome or objective, and any constraints or limitations that may affect the allocation of resources.

## What are some common techniques used to solve the Optimization Rent Problem?

Some common techniques used to solve the Optimization Rent Problem include linear programming, dynamic programming, and genetic algorithms. These techniques involve using mathematical models and algorithms to find the optimal solution.

## What are some real-world applications of the Optimization Rent Problem?

The Optimization Rent Problem has many real-world applications, such as in urban planning, resource management, and production planning. It can also be used in industries such as transportation, agriculture, and finance to optimize the use of resources and maximize profits.

## How can the Optimization Rent Problem benefit businesses and organizations?

Solving the Optimization Rent Problem can help businesses and organizations make more informed and efficient decisions when it comes to allocating resources. This can lead to cost savings, increased profits, and improved overall performance.

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