Layla sells selfmade candles. The asking price has an influence on the total candles being sold and thus also the profit. The expected amount of profit can be calculated with the equation TP = -15p^2 + 195p - 450 . In this equation TP stands for Total Price and the p stands for price in dollars. 


a. How much profit can Layla expect when the price per candle is 4,50 dollars?

b. What would be the coordinates of the points of intersection of the graph of TP with the horizontal axis?

c. What do these points of intersection mean to Layla?

d. With what price is the amount of profit the highest? How much money can Layla make at maximum? 

 Feb 28, 2019

\(TP(p) = -15p^2 + 195p - 450\)


\(a) \text{find }TP(4.5)\)


\(b) \text{at this intersection }TP(p) = 0\\ \text{This is a simple quadratic equation. Solve it for it's roots. }\\ \text{Use the quadratic formula if you have to}\)


\(c)~\text{0 profit means why bother}\)


\(d)~\text{We want to find the maximum value of TP.}\\ \text{As }TP(p) \text{ is a quadratic equation it's graph will be a parabola}\\ \text{In this case as the coefficient of the }p^2 \text{ term is negative it will be facing downwards}\\ \text{and thus the maximum value will occur at the vertex of the parabola}\\ \text{A quick formula for the }x \text{ coordinate of the vertext of }a p^2 + bp + c \\ \text{is }p = -\dfrac{b}{2a} \text{, in this case }p = -\dfrac{195}{2(-15)} = \dfrac{13}{2}\\ \text{Plug this value into }TP(p) \text{ and read off the maximum profit that can be achieved}\)

 Feb 28, 2019

a)   Profit  at 4.5 dollars =   -15(4.5)^2 + 195(4.5) -450 = $123.75


b) The intersection  with the horizontal axis is where

-15p^2 + 195p - 450 = 0      multiply through by -1


15p^2 - 195p + 450 = 0     divide through by 15


p^2 - 13p  + 30   =  0       factor


(p - 10) (p - 3) = 0


Set each factor to 0  and solve for p we get that  p=  3   and p = 10


c)  These points mean that she will earn a profit as long as the price is between $3 and $10


d) As Rom found....p = 13/2 = 6.5 dollars earns the max profit


And this max profit is   -15(6.5)^2 + 195(6.5) - 450 = $183. 75



cool cool cool

 Feb 28, 2019

I really don't understand why you insist on solving all these problems to completion.


Do you want the students to cheat? Because that's what you're doing.  Helping students to cheat.

Rom  Feb 28, 2019

Hi, OP here. I do actually prefer CP "solving" the question. Seeing the steps and how he does it really does help me understand what he's doing so I can administer it to all my other problems that are like this. Not all students come here with their mind set on having other people solve their problems for their own gain, I come here to understand. My native language isn't English and mathematics isn't my forte at all so to be frank I do prefer seeing a simplified version. Thanks for the help though! I do appreciate it a lot :)

Guest Mar 2, 2019

I think that if Rom's answer was not adequate for you that you should have had to ask for mor help yourself.

And you should have been as clear as possible about what you needed help with.


You are doing a course in English and so you need to learn the jargen that goes with it. 

Plus if you struggle a little with the English that is all the more reason why you need to ask further questions yourself.

You need to learn to use these words!

Melody  Mar 2, 2019

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