The weekly demand for DVDs manufactured by a certain media corporation is given by
p = −0.0006x2 + 65
where p denotes the unit price in dollars and x denotes the quantity demanded. The weekly total cost function associated with producing these discs is given by
C(x) = −0.002x2 + 13x + 4000
where C(x) denotes the total cost (in dollars) incurred in pressing x discs. Find the production level that will yield a maximum profit for the manufacturer. Hint: Use the quadratic formula. (Round your answer to the nearest integer.)

Question

11-11-2016
Mathematics

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The weekly demand for DVDs manufactured by a certain media corporation is given by
p = −0.0006x2 + 65
where p denotes the unit price in dollars and x denotes the quantity demanded. The weekly total cost function associated with producing these discs is given by
C(x) = −0.002x2 + 13x + 4000
where C(x) denotes the total cost (in dollars) incurred in pressing x discs. Find the production level that will yield a maximum profit for the manufacturer. Hint: Use the quadratic formula. (Round your answer to the nearest integer.)

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Let the production level that will yield a maximum profit for the manufacturer be x.
The unit price of the disc is given by p = -0.0006x^2 + 65.
The revenue from selling x discs (R(x)) = px = -0.0006x^3 + 65x

Profit = Revenue - Cost = -0.0006x^3 + 65x - (-0.002x^2 + 13x + 4000) = -0.0006x^3 + 0.002x^2 + 52x - 4000

For maximum profit, dP/dx = 0
-0.0018x^2 + 0.004x + 52 = 0
Using quadratic formular, x = 171

Therefore, the production level that will yield a maximum profit for the manufacturer is 171 discs.