Question

The quantity demanded each month of the Walter Serkin recording of Beethoven's Moonlight Sonata, manufactured by Phonola Record Industries, is related to the price/compact disc. The equation
p = -0.00048 x + 7\ \ \ \ \(0<=x<=12,000\)
where p denotes the unit price in dollars and x is the number of discs demanded, relates the demand to the price. The total monthly cost (in dollars) for pressing and packaging x copies of this classical recording is given by
C(x) = 600 + 2x - 0.00003 x^2 \ \ \ \ \(0<=x<=20,000\)
To maximize its profits, how many copies should Phonola produce each month? Hint: The revenue is R(x) = px, and the profit is P(x) = R(x) - C(x). (Round your answer to the nearest whole number.)
discs/month

Answer 1

All these production maximization problems are done the same way. You have the formulas; max/min is fund using the derivative.

we want maximum profit
p(x) = r(x) - c(x)
= x(-.00048x + 7) - (600+2x - .00003x^2)
= -0.00045 x^2 + 5x - 600

max profit occurs where p'(x) = 0, so we solve
-.0009x + 5 = 0
x = 5556

No plug that into p(x) if you want the value of the maximum profit