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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 per compact disc. The equation

p= -0.00051x+6 (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-0.0001x^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.)

Answer 1

Samantha,

 

The hint in the last paragraph tells you how to get the polynomial for the Profit function P(x).  BTW, don't confuse this big P with the small p which is price.  Then, the Profit function is:

 

P(x) = xp(x) - C(x) = -0.00041 x² + 4x - 600

 

Maximum profit occurs when the derivative is set to 0 and solved for x:

 

dP(x)/dx = -0.00082 x + 4 = 0 ,   so x = 4/0.00082 = 4,878 copies each month.