MATHEMATICS · Multi-part · AP · Calculus AB · Analytical Applications of Differentiation · Analytical Applications of Differentiation
Problem
A technology firm sells portable battery packs. When the selling price is p dollars per unit, the weekly demand is modeled by x=300−2p, where 0≤p≤150. The weekly cost of producing x units is given by C(x)=30x+5000 dollars.
(a) Express the weekly profit P as a function of x. Specify the practical domain for x.
(b) Determine all critical points of P on the domain from part (a). Show each step in your calculation.
(c) Identify the production level that yields the maximum weekly profit. Justify your answer by comparing the sign of P′(x) on intervals determined by the critical points and by considering any relevant endpoints.
(d) For the production level found in part (c), calculate the corresponding selling price and the maximum weekly profit. Interpret each of these two values in complete sentences.