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

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=-0.0004x+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

anonymous · Answer

$$Profit = Revenue- Cost$$So,$$P=px-C=(-0.0004x+7)x-(600+2x-0.00001x^2)$$

anonymous · Answer

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...

anonymous · Answer

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).