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

Mathematics

Mclaughlin

30 January, 20:41

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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Answers (1)

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Kianna Briggs · Answer 1

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.