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

Profit = Revenue - Cost

Revenue = R = px = -0.0006x^3 + 55x

Cost = C =-0.004x^2 +14x + 4000

Profit = R-C = -0.0006x^3 +0.004x^2 + 41x -4000

take the derivative and set =0

(R-C)' = -0.0018x^2 + 0.008x + 41 = 0

18x^2 - 80x - 410,000 = 0

9x^2 -40x -205,000=0

x = 40/18 +(1/18)sqr(40^2 +4(9)(205,000))

= (2716.91001+40)/18

= 153.1617

=153 rounded off to nearest integer