maximum profit problem

May 2010
1
0
A company is selling books. they need to know how much to sell each book for max profit. We are given: At $20 per book they sell 17,800. At $15 per book they sell 22,960. Their cost per book is $4.90. What price should they sell at to get max profit?
I found the slope and then linear equation using those numbers (-1/1032). as well as the y-intercept (37.25)

I know i need to now make it a quadratic equation to find the vertex for max profit. ax^2+bx+c and i know a will be negative. But i don't know what numbers to plug into which parts of the equation. Meaning: what is a, b , c and x? which variables represent cost, price, quantity, etc?
i got slope as -1/1032. so i think for every dollar the price decreases they sell 1032 more books. y-intercept is 37.25. the price when there is zero demand for books.
I'm stuck on what numbers to put into quadratic equation now.
please help, thanks
 
Feb 2008
383
38
1st column = price of books
2nd column = no. of books sold. = -1032*1st column+38440
3rd column = cost of books = 4.9*2nd column
4th column = profit = 3rd column - 2nd column*1st column.

(4.9*(-1032*x+38440))-(-1032*x+38440)*x
is the equation to the max profit. after putting 1,2,3,4 together.

after rearranging and simplify.
(4.9*(-1032*x+38440))-(-1032*x+38440)*x
=1032x^2-43496.8x+188356

then to find max. you differentiate the equation to
2064x-43496.8=0
to find x
2064x=43496.8
x= 21.07

hence the max profit = $21

please ask questions if you dont understand as i havent explain it fully.
 

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