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Question

Maximizing Profits

 The weekly demand for DVDs manufactured by a certain media corporation is given by

p = −0.0003x² + 75

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.)

x =   DVDs

Answer 1

Profit P = xp - C, with income xp being the quantity x times price per unit p. After combining terms,

P = -0.0003x³ +0.002x² + 62x -4000. The derivative P' = -.0009x² + 0.004x + 62 --> zero at a max or min.

Quadratic formula becomes x = {-0.004 ± √ [(0.004)² -4(-0.0009)(62)] } / (-0.0018). The [] term is 0.2232, with a square root of ±0.4724. Noting the negative denominator, choose the - sign to make the numerator negative also. Then x = (-0.004 - 0.4724) /(-0.0018) = 265. I verified that this answer is correct by calculating P(265) vs slightly higher and lower values.