Optimization (I believe it's called) word problems

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Homework Help Overview

The discussion revolves around a real estate optimization problem involving rental income from apartments. The scenario describes a company with 180 apartments that are fully occupied at a rent of $300, and how changes in rent affect occupancy and income.

Discussion Character

  • Exploratory, Conceptual clarification, Mathematical reasoning, Problem interpretation

Approaches and Questions Raised

  • Participants explore the relationship between rent increases and occupancy rates, attempting to formulate equations for cost and number of apartments occupied. There are discussions about the nature of the revenue function and its parabolic shape.

Discussion Status

Several participants are engaged in clarifying the equations and variables involved, particularly the definitions of the variables and their implications. There is an ongoing exploration of how to derive a reasonable answer from the established relationships, with some participants expressing confidence in their understanding while others seek further clarification.

Contextual Notes

There are indications of confusion regarding the definitions of variables and the setup of equations, particularly concerning the increments in rent and their effects on occupancy. Participants are encouraged to explore specific revenue calculations for various rent levels to gain insight into the problem.

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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)
 
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UltimateSomni said:

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
 
How do you know it's a parabola?
 
UltimateSomni said:
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
 
UltimateSomni said:
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. :confused:
 
so let's call d a one collar increase

so c=300+10d

what's the other equation then
 
UltimateSomni said:
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.
 
  • #10
so then how do I fix it to get a reasonable answer
 
  • #11
got it, I hope I understand the other ones now
 

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