The quantity demanded each month of the Walter Serkin recording of Beethoven\'s

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 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 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

Explanation / Answer

You have been given the functions p(x) and C(x). If each disc is sold for price p(x) and there are x many discs, the total income from sales, or revenue, is R(x) = x*p(x), as described. The profit is equal to revenue minus cost, or P(x) = R(x) - C(x), again as described. It's a little irresponsible of the writer to use p and P as separate functions, but try to keep them straight. You can write P(x) as a polynomial function of x, and you can simplify it by combining like terms. You should get a second-order polynomial. In order to find the maximum profit, you need to find the maximum point of the function, and this is done by setting the derivative equal to zero. You should know how to take the derivative of a polynomial function, and the derivative of a quadratic function will be linear. You should also be able to solve the linear equation (set equal to zero) for x, giving you the number of copies that should be produced each month. If p(x) is only defined for 0

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