rent problem solving- tough!!?

Question

The manager of a large apartment complex knows from experience that 100 units will be occupied if the rent is 420 dollars per month. A market survey suggests that, on the average, one additional unit will remain vacant for each 7 dollar increase in rent. Similarly, one additional unit will be occupied for each 7 dollar decrease in rent.

a. Give an expression for the monthly revenue in terms of x, the monthly rent being charged.

b. What rent should the manager charge to maximize revenue?

Answer 1

So we know that when x = 420, the number of occupied apartments (call it n) = 100 and the revenue (r) = 420*100 = 42000.

Now when x decreases by 7, n increases by 1. So if we're trying to write an equation in the form y = mx + b, then we could start by saying n = (-1/7)x + b.

We can find b by plugging in the fact that n = 100 when x = 420. So 100 = (-1/7)420 + b. (-1/7)420 = -60, so 100 = -60 + b, so b = 160.

So now we have n = -x/7 + 160. We can check this by plugging in some numbers, such as the original x = 420: n = -60 + 160 = 100. If we increase the rent by 7, so x = 427, n = -61 + 160 = 99--sure enough, one tenant has left.

But we need the equation to give r, th revenue, which is n*x. So, n*x = ((-1/7)x + 160)x. So r = -x^2/7 + 160x. That's the answer to the first part of the question.

Now for the second part, we need to use some calculus. Maxima and minima can only be found at the endpoints or when the derivative of the function is 0. In this case, the endpoints correspont to charging no rent or having no tenants, and are therefore obviously minima. So, we need to find where the derivative is 0.

The original equation was r = -x^2/7 + 160x. The derivative of this is r' = -2x/7 + 160. For that to be equal to 0, 2x/7 sould be equal to 160. So multiply both sides by 7 and 2x = 1120. Then divide by 2 and x = 560. That's the rent the manager should charge.

Answers:

a) r = -x^2/7 + 160x.
b) $560.

Answer 2

Assume that the complex has only 100 units available for rent. All the units are rented at $420.

Revenue is the rent charged times the number of occupied units.

Occupancy rates are 100 - (# of $7 rent increases)
Rent is $420 + 7 * (# of $7 rent increases)
Maximize your Revenue by finding the maximum Occupancy rate * Rent

The rest is left for the student....
Just so you can check: Looks like the best rent to charge would be $560

Answer 3

Let U(x) be the number of units rented when the rent is x.
We know that U(420 + 7y) = 100 - y
We want to find U(x), so let:
x = 420 + 7y
7y = x - 420
y = x/7 - 60
So U(x) = 100 - (x/7 - 60) = 160 - x/7

The total revenue collected is the rent times the number of units rented or:

a) Revenue R = x * U(x) = x(160 - x/7) =
160x - x^2/7

b) Take the derivative of revenue with respect to x and set equal to zero to find the rent x which maximizes revenue.

dR/dx = 160 - 2x/7
160 - 2x/7 = 0
2x/7 = 160
x = 560

Answer 4

Let n = number of additional units occupied
Let x = monthly rent per unit
f(x) = total monthly revenue

a. It is easier to initially set up the total monthly revenue equation in terms of n, then substitute in (x = 420+7n) to get it in terms of x.

In terms of n: f(n) = (100-n) * (420+7n)
In terms of x: f(x) = [100 - (x-420)/7] * x = 160x - (1/7)x^2

b. Take the derivative and set equal to zero to find the maximum point on the total monthly revenue curve.
f(x) = 160x - (1/7)x^2
f'(x) = 0 = 160 - (2/7)x
x = 560

Edit: I'm not sure why all the thumbs down. This answer is correct, as is Zaphod's and Amy F's.

Answer 5

How many units in the apartment property? You don't give the upside of that. All I know is that 100 units are occupied with the rent at $420 a month. Are there more units or not? Yes, it will change the equation.

Answer 6

a. Pay the rent motherf**ker. Where's my $x.

b. The rent should be $1200 a month. $420 won't cover taxes and expenses these days. You can lose half your tennants and still come out ahead with lower upkeep costs.

Ignore the market survey. It doesn't matter. It's never accurate. Don't take it as an absolute.

I solved it with common sense instead of math.
If this is for your homework I'm no help at all.
Tell your teacher math is overrated.
Common sense trumps math.

Answer 7

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Answer 8

i say charge 399 a month, rent out 3 more apartments, and be happy.

if $420/month=100 units,
and 427/month=99 units
and 399/month = 103 units,

then lets see, x=rent, y = units

x=100 units
x+7=99
x-7=101

i'm not sure what you are looking for

Answer 9

Finally! A practical application math question!

I have no clue, but I"m pretty sure an "x-7" and "x+7" figure into the equations.

Answer 10

get someone else to do ur home work!