Minimizing average cost

Question

A wood pellet factory can produce x hundred units of wood pellets per day at a daily cost of C(x)=0.6x^2+21x+1215, in dollars.

a. find the marginal cost

b. Show that, when the average cost is at a minimum, the average cost and the marginal cost are equal.

Answer 1

C(x) = 0.6x² + 21x + 1215

Marginal cost = C'(x) = 1.2x + 21

Average cost = A(x) = C(x) / x = 0.6x + 21 + 1215 / x

A'(x) = 0.6 - 1215 / x² = (0.6x² - 1215) / x² = 0

0.6x² = 1215

x² = 2025 So, x = 45

Minimum average cost when x = 45

At x = 45, marginal cost = C'(45) = 1.2(45) + 21 = 75

Minimum average cost = A(45) = 0.6(45) + 21 + 1215/45 = 75