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?

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jlmpco jlmpco · Answer 1 · 2019-11-08T04:59:52+01:00

Answer:

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