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Question 1201982: the cost of making x thousand pretzels per week is c=.4x^2-28x+87 and the revenue from selling x thousand pretzels per week is r=-.6x^2+8x find the equation for profit where (p=r-c) C and R are in thousands of dollars
Answer by math_tutor2020(3817)   (Show Source):
You can put this solution on YOUR website!
Answer: P = -x^2 + 36x - 87
Work Shown:
Profit = Revenue - Cost
P = R - C
P = (-0.6x^2 + 8x) - (0.4x^2 - 28x + 87)
P = -0.6x^2 + 8x - 0.4x^2 + 28x - 87
P = (-0.6x^2-0.4x^2) + (8x+28x) - 87
P = -x^2 + 36x - 87
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Edit: The student requested finding the vertex.
There are few ways to find the vertex. One way is to note that the parabola y = -x^2+36x-87 is of the form y = ax^2+bx+c
a = -1
b = 36
c = -87
The vertex is of the form (h,k)
The steps to calculate h would be:
h = -b/(2a)
h = -36/(2*(-1))
h = -36/(-2)
h = 18
This is the x coordinate of the vertex.
Plug that into the function to find its paired y coordinate.
y = -x^2+36x-87
y = -18^2+36(18)-87
y = 237
The vertex is located at (h,k) = (18, 237)
It means that selling 18 thousand pretzels will bring in a max profit of 237 thousand dollars.
The point (18, 237) is the highest point on the parabola.
I recommend using a graphing app such as Desmos or GeoGebra to confirm the answer is correct.
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