Optimization (I believe it's called) word problems

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In summary, the real estate company owns 180 apartments that are fully occupied at a rent of $300. For every $10 increase in rent, 5 apartments become unoccupied. The goal is to find the rent that will generate maximum income for the company. Using the equations c = 300 + 10d and a = 180 - 5d, we can determine the rental income for different values of d (representing the number of $10 increases). It is important to note that the number of apartments, a, must always be positive. Therefore, the maximum rental income is achieved when d = 6, which results in a rent of $360 and a total of 150 occupied apartments. Any further increase in rent will
  • #1
UltimateSomni
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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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  • #2
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.
 
  • #3
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
 
  • #5
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.
 
  • #6
d is a 10 dollar increase. d=10 is a 100 dollar increase
 
  • #7
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:
 
  • #8
so let's call d a one collar increase

so c=300+10d

what's the other equation then
 
  • #9
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
 

1. What is optimization in word problems?

Optimization in word problems refers to the process of finding the best solution or outcome for a given scenario, subject to certain constraints or limitations. It involves using mathematical techniques to determine the optimal value of a variable that will result in the most desirable outcome.

2. What are some common examples of optimization word problems?

Some common examples of optimization word problems include maximizing profit, minimizing cost, finding the shortest or fastest route, and determining the optimal production level for a given resource.

3. How do you approach solving an optimization word problem?

To solve an optimization word problem, you must first identify the objective or goal, the constraints or limitations, and the variables involved. Then, you can use mathematical techniques such as calculus, linear programming, or the simplex method to find the optimal value for the variable.

4. What are some challenges or difficulties in solving optimization word problems?

One of the main challenges in solving optimization word problems is determining the appropriate mathematical model to use. It can also be challenging to accurately represent and interpret real-world scenarios in mathematical equations. Additionally, finding the optimal solution may require multiple iterations and can be time-consuming.

5. How can optimization word problems be applied in real-world situations?

Optimization word problems have a wide range of applications in fields such as economics, engineering, business, and science. They can be used to optimize production processes, minimize costs, improve efficiency, and make informed decisions in various industries and organizations.

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