Mathematics

Problem

An online retailer sells a limited-edition gadget at a fixed price of \

80 per unit. The total cost (in dollars) of producing

q$ units per month is modeled by

C(q)=10\,000+30q+0.01q^{2}.

Assume the retailer can sell every unit produced. (a) Derive the monthly profit function

P(q)

in terms of

q

. (b) Determine the production level

q

that maximizes monthly profit. Show all necessary calculus steps. (c) Compute the maximum monthly profit, rounding to the nearest dollar.