Find Profit Function P(x)

label Mathematics
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The revenue and cost functions are given.

R(x)=20x-0.1x^2

C(x=4x+2)

Where x=number of staplers, R(x) and C(x) are in Dollars.

A. Find the profit function P(x)

B. Find the # of staplers which need to be sold in order to maximize profit.

C. Find the maximum profit.

Oct 1st, 2015

Hello!

A. P(x) = R(x) - C(x) = 20x - 0.1*x^2 - 4x - 2 = -0.1*x^2 + 16x - 2.

B. This P(x) is a parabola branches down. It has a maximum at x = -b/(2a)
where a is a factor at x^2 (here -0.1) and b is a factor at x (here 16).

So the answer for B is -16/(2*(-0.1)) = (16/2)*10 = 80 (staplers).

C. The maximum profit is P(x) where x is from B.

P(80) = -0.1*(80)^2 + 16*80 - 2 = -640 + 1280 - 2 = 640 - 2 = 638 (dollars).

Please ask if something is unclear.
Oct 1st, 2015

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