An outdoor-gear manufacturer markets an insulated water bottle
MATHEMATICS · Multi-part · AP · Calculus AB · Analytical Applications of Differentiation · Analytical Applications of Differentiation
Problem
An outdoor-gear manufacturer markets an insulated water bottle. Market data indicate that when the weekly production level is q thousand bottles (q≥0), the corresponding wholesale price in dollars is given by
p(q)=50−0.4q.
The total weekly cost (in dollars) of producing q thousand bottles is modeled by
C(q)=4000+8q+0.01q2.
Assume the manufacturer can meet any level of demand generated by this price function.
(a) Write the weekly revenue R(q) (in dollars) as a function of q. Clearly state the domain that is relevant for this context.
(b) Define the weekly profit P(q) in terms of q and write it explicitly.
(c) Determine the production level q (in thousands of bottles) that maximizes weekly profit. Show all analytical work, including any derivatives used.
(d) Verify that the production level found in part (c) yields a maximum, not a minimum, for the profit function. Justify your conclusion analytically.
(e) For the optimal production level found in part (c), calculate the corresponding wholesale price and the maximum weekly profit, rounding each monetary value to the nearest dollar.