A firm producing electric bicycles faces a demand relationship given by where is the…
MATHEMATICS · Multi-part · AP · Calculus AB · Analytical Applications of Differentiation · Analytical Applications of Differentiation
Problem
A firm producing electric bicycles faces a demand relationship given by p(q)=1,200−6q, where q is the number of bicycles sold per week and p is the selling price in dollars per bicycle. The weekly cost of producing q bicycles is modeled by
C(q)=100q2+30,000,
where C(q) is measured in dollars.
The firm’s weekly profit is P(q)=R(q)−C(q), where R(q) is the weekly revenue.
Assume the production capacity of the factory is between 0 and 160 bicycles per week.
(a) Write an explicit formula for the revenue function R(q) solely in terms of q. Do not simplify numerical coefficients.
(b) Derive an expression for the marginal profit P′(q). Show algebraic work leading to your result.
(c) Determine all production levels in the interval 0<q<160 at which profit is stationary. Classify each stationary point as yielding a local maximum, a local minimum, or neither. Justify your answer analytically.
(d) Identify the production level that maximizes weekly profit within the stated interval, and calculate the corresponding maximum profit. Interpret your result in the context of the business.