An online retailer sells a limited-edition gadget at a fixed price of 80 dollars per unit. The total monthly cost, in dollars, of producing q units is C(q)=10000+30q+0.01q2. 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.