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

Maximizing Profits

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

p = −0.0006x² + 55

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) = −0.001x² + 14x + 4000

where C(x) denotes the total cost (in dollars) incurred in pressing x discs.

Answer 1

Revnue = px = -0.0006x^3 +55x

Profit = Revenue -Cost = -0.0006x^3 + 55x + 0.001x^2 -14x -4000

Profit maximizing output level is found by setting the derivative = 0,

P' = R'-C' = -0.0018x^2 +0.002x +41=0

use the quadratic formula, x=-b/2a + or - (1/2a)sqr(b^2-4ac)

x = 0.002/0.0036 + or - (1/0.0036)sqr(0.002^2 +4(0.0018)(41)) = 151.5 per week

Max P = -0.0006(151.5)^3 +0.001(151.5)^2 + 41(151.5) -4000= -2063.4+2211.5 = $148.10

price = p=-0006(151.5)^2+55 = $41.20