Textbook Problem

Suppose that the marginal cost for a certain product is given by $\overline{\text{MC}}=1.05{(x+180)}^{0.05}$ and marginal revenue is given by $\overline{\text{MC}}=\left(1/\sqrt{0.5x+4}\right)+208,$ where x is in thousands of units and revenue and cost are in thousands of dollars. Suppose further that fixed costs are $150,000 and production is limited to at most 200 thousand units.

(a) Find C(x) and R(x).

(b) Graph C(x) and R(x) to determine whether a profit can be made.

(c) Determine what level of production yields maximum profit and find the maximum profit (or minimum loss).