Question

The quantity demanded each month of the walter serkin recording of beethoven's moonlight sonata, manufactured by phonola media, is related to the price per compact disc. the equation p = −0.00054x + 9 (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 − 0.00002x2 (0 ≤ x ≤ 20,000)

Answer 1

We can get the Profit function P(x) from the Hint. 

the Profit function is: P(x) = xp(x) - C(x) = -0.00041 x2 + 4x - 600
Attention: don't get confuse by the big P of the profit with the small p of the price To calculate the maximum profit, we need to find the derivative of P(x) then set it to 0 then find x: dP(x)/dx = -0.00082 x + 4 = 0 ,   so x = 4/0.00082 = 4,878 copies each month.