SOLUTION: A manufacturer can sell headphones at a price of 140- .01x dollars each. It costs 40x + 15,000 dollars to produce all of them. How many headphones should be produced to maximize pr

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Question 1056801: A manufacturer can sell headphones at a price of 140- .01x dollars each. It costs 40x + 15,000 dollars to produce all of them. How many headphones should be produced to maximize profit? (application of derivative)
Answer by solve_for_x(190) About Me  (Show Source):
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Profit is the difference between revenue and cost.

With a price of p = 140 - 0.01x, the sale of x headphones gives revenue of:

R(x) = px = (140 - 0.01x)x = 140x - 0.01x^2

The costs are:

C(x) = 40x + 15000

The profit is then:

P(x) = R(x) - C(x) = (140x - 0.01x^2) - (40x + 15000)

P(x) = -0.01x^2 + 140x - 40x - 15000

P(x) = -0.01x^2 + 100x - 15000

Taking the derivative of P(x) gives:

P'(x) = -0.01(2)x + 100

P'(x) = -0.02x + 100

Setting the derivative equal to 0 and solving for x then gives:

P'(x) = -0.02x + 100 = 0

0.02x = 100

x = 100 / 0.02

x = 5000

The maximum profit corresponds to the production of 5000 headphones.