Maximizing Profit for a 50-Unit Apartment Complex

[mathkid3]

A real Estate office handles a 50-unit apt. complex. When the rent is \$580/mo. all units are occupied.

For each \$40 increase in rent, however, an avg of one unit becomes vacant. Each occupied unit requires an avg of \$45 per month for service and repairs.

What rent should be charged to obtain a maximum profit?
I need help setting up the problem

TY!

 

[MarkFL]

The dollar sign character is used for rendering LaTeX here, that is why your post looks that way.

We need to find a function which describes the number of units occupied as a function of the rent charged. Let the rent be $\displaystyle 580+40x$

We also know that for each increase of 1 in x, we get a decrease of 1 in U, the number of units occupied. Hence:

$\displaystyle \frac{dU}{dx}=-1$ where $\displaystyle U(0)=50$

Can you now solve this IVP to find $\displaystyle U(x)$?

You may choose to simply use the point-slope formula to find this linear function.

 

[mathkid3]

I may need a little further assistance in what to differentiate

first one I have seen like this

 

[MarkFL]

First, we want to find the number of units occupied as a function of the number of 40 dollar increases in rent.

So, either integrate the ODE I gave, or more simply use the point-slope formula to find the function. You know a point (0,50) and the slope m = -1.

 

[mathkid3]

im getting y=-x+50now what coach ? :)

 

[MarkFL]

Okay, so we now have:

$\displaystyle U(x)=50-x$

Now, we know profit is revenue minus cost. Can you state the profit as a function of U and x?

$\displaystyle P(U,x)=?$

Note: don't worry, we will get rid of U to get the profit as a function of one variable in the next step.

 

[mathkid3]

Im sorry Mark I don't follow

 

[MarkFL]

For the revenue:

The money coming in is the number of units occupied times the rent charged per unit.

What is this product?

For the cost:

The total cost is the number of units occupied times the cost of upkeep per unit.

What is this product?

 

[mathkid3]

P = 580(50)-50(45)?

 

[MarkFL]

That's only true if x = 0. Recall we defined the rent as:

$\displaystyle r(x)=580+40x$

And instead of using 50 for the units occupied, we want to use $\displaystyle U(x)$.

What do you get now?

 

[mathkid3]

hey is the profile pic u?

P = (580+40z)(50)(45) ?

 

[MarkFL]

No, that is one of my intellectual heroes, Dr. Ed Witten, the only physicist to win the Fields Medal in mathematics, and arguably the world's foremost expert in M-theory.

Now, revenue is number of units occupied ($\displaystyle U(x)=50-x$) times rent charged ($\displaystyle r(x)=580+40x$), so we have the revenue:

$\displaystyle R(x)=(50-x)(580+40x)$

And cost is units occupied ($\displaystyle U(x)=50-x$) times average cost per unit (45 dollars), so we have the cost function:

$\displaystyle C(x)=45(50-x)$

So, what is the profit function $\displaystyle P(x)$?

 

[mathkid3]

P = (50-z)(580+40z)-45(50-z)?

 

[MarkFL]

Yep, now I would factor, expand, then optimize. You don't have to expand, you could use the product rule to differentiate. Your choice.

edit: What kind of number do we require x to be?

 

[mathkid3]

P ' = -40z^2 + 1465z + 26750?

am I done?

 

[MarkFL]

That's not the derivative, that's the expanded profit function. Now differentiate and equate to zero...but be careful as z must be what kind of number? Hint: look at the function defining the number of units occupied.

 

[mathkid3]

wait is Z = -1 ?

 

[MarkFL]

No, but how did you determine that value?

 

[mathkid3]

I was just thinking by the way u said it that z = -1is it that z must be positive ?

 

[MarkFL]

First, think about the fact that U must be a non-negative integer in [0,50], so what kind of number must z be and what is the feasible domain?

 

[mathkid3]

Mark,

sorry I fell asleep on couch last night while chatting with you on the forum. I had to get coffee in me to start to re what we were doing last night :)

ok I have questions. #1 how did I get so lost in the problem from the start?? I think it was from asking your help you introduced so variables in the equation that didn't match with the textbook. Prob just overwhelmed me from the start.

I left u with giving you the expanded profit function instead of taking the derV and setting to zero

let me try again...

P ' (x) = -80x + 14652

solving the org P function for x I get x =
-1387.75 and x = -1452.25 This is for defining the feasible domain (Is that correct Mark?)

anyway... I wish we could start from scratch in this problem and u could walk me through it again. I do not like it when I get lost in a problem. I need to understand why we set it up the way we did and why I am struggling with a concept(s)

Thanks bud!

 

[MarkFL]

Okay, we are given:

A real Estate office handles a 50-unit apt. complex. When the rent is \$580/mo. all units are occupied.

For each \$40 increase in rent, however, an avg of one unit becomes vacant. Each occupied unit requires an avg of \$45 per month for service and repairs.

What rent should be charged to obtain a maximum profit?

Here's the way I would look at it:

Let z represent the number of \$40 increases in rent, that is, the monthly rent per unit is:

$\displaystyle r(z)=580+40z$

We are told that for each increase of 1 in z, there is a decrease of 1 in the number of units occupied, which we can state as:

$\displaystyle u(z)=50-z$

The monthly revenue is the total amount of rent collected, which is equal to the number of units occupied times the monthly rent per unit:

$\displaystyle R(z)=u(z)\cdot r(z)=(50-z)(580+40z)=20(50-z)(29+2z)$

The total monthly cost is the average cost per unit times the number of units occupied:

$\displaystyle C(z)=45u(z)=45(50-z)$

Now, the monthly profit is the monthly revenue minus the monthly cost:

$\displaystyle P(z)=R(z)-C(z)=20(50-z)(29+2z)-45(50-z)=5(50-z)(4(29+2z)-9)=5(50-z)(107+8z)$

Now, I am going to let you use differentiation, while I am going to use the fact that we have a parabolic profit function opening downward whose axis of symmetry will be midway between the roots, and thus the vertex (maximum point) will be on this axis.

The roots are:

$\displaystyle 107+8z=0\,\therefore\,z=-\frac{107}{8}$

$\displaystyle 50-z=0\,\therefore\,z=50$

The axis of symmetry is then the line (using the mid-point formula):

$\displaystyle z=\frac{-\frac{107}{8}+50}{2}=\frac{293}{16}$

See if you can get this critical value using the calculus. After this, we'll deal with the fact that we should treat z as a discrete variable rather than a continuous one.