If C(x) is the cost of producing x units of a commodity, then the average cost per unit is c(x) = C(x)/x. Consider the cost function C(x) given below. C(x) = 81,000 + 270x + 6x^(3/2)

Question

(a) Find the total cost at a production level of 1000 units. (Round your answer to the nearest cent.)

(b) Find the average cost at a production level of 1000 units. (Round your answer to the nearest cent.)

(c) Find the marginal cost at a production level of 1000 units. (Round your answer to the nearest cent.)

(d) What is the minimum average cost? (Round your answer to the nearest dollar.)

Answer 1

a) Total cost C(1000) = 81,000 + 270,000 + 6 • 31,622.777 = $540,736.66

b) Average cost c(1000) = C(1000)/1000 = $540.74

c) Marginal cost is first derivative of C: C'(x) = 270 + 9 • x^0.5 and C'(1000) = $554.60

d) Average cost c(x) = 81,000/x + 270 + 6 • x^0.5 and c'(x) = -81,000 • x^-2 + 3 • x^-0.5. Set c' = 0 to find max or min of c(x): 81,000 / x² = 3 / x^0.5 yielding 27,000 = x^1.5 and x = 900. The second derivative c"(900) is positive, confirming that this is a minimum. Finally c(900) = 81,000/900 + 270 + 6 • 30 = $540.