# Optimization (I believe it's called) word problems

## Homework Statement

A real estate company owns 180 apartments, which are fully occupied when the rent is $300. The company estimates that for each$10 increase in rent, 5 apartments will become unoccupied. What rent should be charged so that the company will receive the max income?

## Homework Equations

x=-b/2a (in ax^2+bx+c)

## The Attempt at a Solution

c=cost
a=apartments
d= 10 dollar increase in price

c=300+d
a=180-5d

multiplying those together gets a nonsense negative (can't decrease the price, the apartments are full)

Mark44
Mentor

## Homework Statement

A real estate company owns 180 apartments, which are fully occupied when the rent is $300. The company estimates that for each$10 increase in rent, 5 apartments will become unoccupied. What rent should be charged so that the company will receive the max income?

## Homework Equations

x=-b/2a (in ax^2+bx+c)

## The Attempt at a Solution

c=cost
a=apartments
d= 10 dollar increase in price

c=300+d
a=180-5d

multiplying those together gets a nonsense negative (can't decrease the price, the apartments are full)
Before trying to answer this question, see if you can figure out what the rental income will be for a few sets of values.

For example, what is the total revenue when they charge $300 per apartment? What is the revenue if they raise the rent by$10?
What is the revenue if they raise the rent by $20? After you get a feel for how this is working, we'll look at what happens when they raise the rent by$x.

I all ready know what the answers is. It increases until 330 (30 dollars increase) and then begins to decrease. It's a parabola from 300 to 360. But I just can't find a way beyond trying numbers to solve it

Mark44
Mentor
How do you know it's a parabola?

eumyang
Homework Helper
c=cost
a=apartments
d= 10 dollar increase in price

c=300+d
a=180-5d

multiplying those together gets a nonsense negative (can't decrease the price, the apartments are full)

d isn't defined well, so the 1st equation (bolded) isn't correct. The way you wrote the equations, if d = 10, then a = 130, not 175. You'll need to rework the 1st equation.

d is a 10 dollar increase. d=10 is a 100 dollar increase

eumyang
Homework Helper
d is a 10 dollar increase. d=10 is a 100 dollar increase

But when you put d = 10 into the equation
c = 300 + d,
you get c = 310, not 400.

so let's call d a one collar increase

so c=300+10d

what's the other equation then

eumyang
Homework Helper
so let's call d a one collar increase

so c=300+10d

what's the other equation then

I'd rather call d, "the number of $10 increases." So d = 10 means that there are 10$10 increases, or a total increase of \$100. Your 2nd equation (for a) is correct.

so then how do I fix it to get a reasonable answer

got it, I hope I understand the other ones now