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=-0.0004x+7 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 xcopies of this classical recording is given by C(x)=600+2x-0.00001x^2 To maximize its profits, how many copies should Phonola produce each month? Hint: The revenue is R(x) = px, and the profi

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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=-0.0004x+7 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 xcopies of this classical recording is given by C(x)=600+2x-0.00001x^2 To maximize its profits, how many copies should Phonola produce each month? Hint: The revenue is R(x) = px, and the profi

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\[Profit = Revenue- Cost\]So,\[P=px-C=(-0.0004x+7)x-(600+2x-0.00001x^2)\]
You expand the right-hand side out and take the derivative with respect to x. You get\[P'(x)=5-0.00078x\]Extrema within the domain occur when the derivative is zero. SO set P'(x)=0 and solve for x. You should get 6410.25641...
You can argue this yields a maximum since the coefficient of the quadratic of the profit function, P, is negative (the quadratic is concave down).

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