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) = 54,000 + 240x + 4x3/2

(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?

a) 420,491.1 b) 420.49 c) 429.68 d) 900 e) 420

Step-by-step explanation:

A) Replace x = 1000 in the cost function

C (1000) = 54,000 + 240*1000 + 4*1000^ (3/2)

C (1000) = 420,491.1

B) Average cost per unit is c (x) = C (x) / x

420,491.1/1000 = 420.49

C) Marginal cost is the additional cost to produce a unit, so we find the cost difference between producing 999 and 1000 units

C (999) = 54,000 + 240*999 + 4*999^ (3/2)

C (999) = 420,061.41

Marginal cost = 420,491.1 - 420,061.41 = 429.68

D) To find the minimum value for units, we find the derivate of average cost function and equal to zero

C (x) / x = 54000/x + 240*x/x + 4*x^ (3/2) / x

C (x) / x = 54000*x^ (-1) + 240 + 4*x^ (1/2)

C' (x) / x = - 54000*x^ (-2) + 4 * (1/2) * x^ (-1/2)

C' (x) / x = - 54000/x^ (2) + 2/x^ (1/2)

Equal to zero

-54000/x^ (2) + 2/x^ (1/2) = 0

2/x^ (1/2) = 54000/x^ (2)

x^ (2) / x^ (1/2) = 54000/2

x^ (3/2) = 27000

x = 900

E) The minimun average cost is obtained replacing 900 in the function cost

C (900) / 900 = (54,000 + 240*900 + 4*900^ (3/2)) / 900

C (900) / 900 = 378000/900 = 420