If C(x) is the cost of producing x units of a commodity, then the average cost p
ID: 3190437 • Letter: I
Question
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. (Round your answers to the nearest cent.) C(x) = 24,000 + 160x + 6x^(3/2) (a) Find the total cost at a production level of 1000 units. (b) Find the average cost at a production level of 1000 units. (c) Find the marginal cost at a production level of 1000 units. (d) Find the production level that will minimize the average cost. (e) What is the minimum average cost?Explanation / Answer
C(x) = 24000 + 160x + 6x3/2
(a) C(1000) = 24000 + 160(1000) + 6(1000)3/2
= 24000 + 1600000 + 189736.66
C(1000) = $373736.66
(b) C(1000)/1000 = 373736.66/1000 = $373.74
(c) Marginal cost = dC/dx = 160 + 6(3/2)x1/2 = 160 + 9(1000) = $444.60
(d) d/dx[C(x)/x] = 0
C(x)/x = 24000/x + 160 + 6x1/2
d/dx[C(x)/x] = - 24000/x2 + 0 + 6(1/2)/x = 0
24000/x2 = 3/x
24000x = 3x2
8000x = x2
64000000x = x4
64000000 = x3
x3 - 64000000 = 0
(x - 400)(x2 + 400x + 160000) = 0
x = 400 is the only real critical value. The roots of the quadratic are complex.
400 units minimizes average cost.
(e) C(400)/400 = 24000/400 + 160 + 6400
= 60 + 160 + 6(20) = 220 + 120 = 340 units