Applications with polynomials

[Gr8fu13]

Hello,
This question analyzes the profit found for sales of decorative tiles. It is a demand question. Question is as follows:

Part 1- Suppose a market research company finds that at a price of p = $20, they would sell x = 42 tiles each month. If they lower the price to p = $10, then more people would purchase the tile, and they can expect to sell x = 52 tiles in a month’s time. Find the equation of the line for the demand equation. Write your answer in the form p = mx + b. Hint: Write an equation using two points in the form (x,p).

Part 2- A company’s revenue is the amount of money that comes in from sales, before business costs are subtracted. For a single product, you can find the revenue by multiplying the quantity of the product sold, x, by the demand equation, p.

Substitute the result you found from part a. into the equation R = xp to find the revenue equation. Provide your answer in simplified form.

Any help with this would be greatly appreciated! I have several other questions that depend on the answers from this question so if I get this wrong, all preceeding questions would be wrong as well. I was figuring it would be something like (xp+12)(xp+12) or something. I may be COMPLETELY off on this. Thanks!

 

[galactus]

Part 1- Suppose a market research company finds that at a price of p = $20, they would sell x = 42 tiles each month. If they lower the price to p = $10, then more people would purchase the tile, and they can expect to sell x = 52 tiles in a month’s time. Find the equation of the line for the demand equation. Write your answer in the form p = mx + b. Hint: Write an equation using two points in the form (x,p).

This is just finding the equation of a line using two points. (20,42), (10,52). Use these points to find the line equation.

First, find your slope. Know how to find a line equation given two points?.

Part 2- A company’s revenue is the amount of money that comes in from sales, before business costs are subtracted. For a single product, you can find the revenue by multiplying the quantity of the product sold, x, by the demand equation, p.
Substitute the result you found from part a. into the equation R = xp to find the revenue equation. Provide your answer in simplified form.

The line equation from part 1 gets subbed in for p.

 

[Gr8fu13]

For part one I came up with p=-1x+62. Does this look right? Can I solve this any further? So the answer for part one would be p=-1x+62?

The second part would look like this: R=x(-1x+62) which would be simplified as R=-1x^2 + 62x? Or would I answer it as R= 61x^2? I REALLY appreciate your help with this!

 

[Deleted member 4993]

For part one I came up with p=-1x+62. Does this look right? Can I solve this any further? So the answer for part one would be p=-1x+62?

You should be able answer that yourself! Check the initial and final conditions you stated.

When x =20 what is the value of p from your equation?

When x =10 what is the value of p from your equation?

If they match your given condition (values of 'p') - you must be right.

If not - you have made some mistake somewhere!

So what do you get?




The second part would look like this: R=x(-1x+62) which would be simplified as R=-1x^2 + 62x? ? Correct



Or would I answer it as R= 61x^2 ? No you cannot do that!!


I REALLY appreciate your help with this!

 

[Gr8fu13]

Okay, I am right thus far. I have answered other questions in between, but now I am stuck on another one. My final equation to represent the profit looks like this P= -1x^2 +56x-300. The question I am stuck on says,"Use trial and error to find the quantity of tile sets per month that yeilds the highest profit" X represents the number of tile sets. I would think that this answer would be infinity. Obviously the more you sell the more profit you gain.

However, the next question asks how much profit I would earn from the number I found in part 1 and what price would I sell the tile sets at to realize this profit. Do you understand this at all?

 

[Mrspi]

Okay, I am right thus far. I have answered other questions in between, but now I am stuck on another one. My final equation to represent the profit looks like this P= -1x^2 +56x-300. The question I am stuck on says,"Use trial and error to find the quantity of tile sets per month that yeilds the highest profit" X represents the number of tile sets. I would think that this answer would be infinity. Obviously the more you sell the more profit you gain.<----that might seem like an obvious conclusion....but it COULD be an incorrect conclusion!

You may want to try something....

pick some numbers to use for x, the number of sets sold.

Try these.....
If x = 10, what is P?

If x = 20, what is P?

If x = 30, what is P?

If x = 100, what is P?

If x = 1000, what is P?

Does it appear that very large sales would produce very large profits?

You might try plotting the points you've determined for the various values of x I suggested. Does that give you a clue as to where you should look for the number of tile sets which will produce the greatest profit?

 

[Deleted member 4993]

Okay, I am right thus far. I have answered other questions in between, but now I am stuck on another one. My final equation to represent the profit looks like this P= -1x^2 +56x-300. The question I am stuck on says,"Use trial and error to find the quantity of tile sets per month that yeilds the highest profit" X represents the number of tile sets. I would think that this answer would be infinity. Obviously the more you sell the more profit you gain.

No - if you plot P(x) = -1x^2 +56x-300 you will see that P is actually an up-side down parabola - that is for a while P increases as x increase but then it flattens out and subsequently starts to decrease for large x . For example for x = 50 ? P = 0 and P<0 for x>50.

So - plot the profit function and look at it various ways.

However, the next question asks how much profit I would earn from the number I found in part 1 and what price would I sell the tile sets at to realize this profit. Do you understand this at all?

 

[Gr8fu13]

okay, I did some substituting and you are correct. The most profit would be from selling 28 sets and after that it decreases. Thanks for that help!! 50 would be the number that would break even, If I hadn't figured that out I would have never gotten part 1 because the proceeding question was how many need to be sold to break even. I appreciate everyone's help, I couldn't have done it without you pointing me in the right direction. I will definitely be back. I love this site! You didn't just give me the answer, you pointed me in the right direction so that I could understand as I was doing it. Thanks again! :wink:

 

[Deleted member 4993]

okay, I did some substituting and you are correct. The most profit would be from selling 28 sets and after that it decreases. Thanks for that help!!

50 would be the number that would break even,

No - the break-even number is profit becomes ?0 for the lowest positive number of articles sold

That number would be 6 for your problem

Mrs? and I DID suggest that you plot this function - did you???

If I hadn't figured that out I would have never gotten part 1 because the proceeding question was how many need to be sold to break even. I appreciate everyone's help, I couldn't have done it without you pointing me in the right direction. I will definitely be back. I love this site! You didn't just give me the answer, you pointed me in the right direction so that I could understand as I was doing it. Thanks again! :wink: