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.

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

Required:
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?

Answer 1

a. C(1,000) = 420,491.11

b. c(1,000) = 420.49

c. dC/dx(1,000) = 429.72

d. x = 900

e. c(900) = 420

Explanation:

We have a cost function for x units written as:

`C(x) = 54,000 + 240x + 4x^(3/2)`

a. The total cost for x=1000 units is:

`C(1,000) = 54,000 + 240(1,000) + 4(1,000)^(3/2)\\\\C(1,000)=54,000+240,000+4\cdot 31,622.78\\\\C(1,000)=54,000+240,000+ 126,491.11 \\\\C(1,000)= 420,491.11`

b. The average cost c(x) can be calculated dividing the total cost by the amount of units:

`c(1,000)=(C(1,000))/(1,000)=( 420,491.11 )/(1,000)= 420.49`

c. The marginal cost can be calculated as the first derivative of the cost function:

`(dC)/(dx)=240(1)+4(3/2)x^(3/2-1)=240+6x^(1/2)\\\\\\(dC)/(dx)(1,000)=240+6(1,000)^(1/2)=240+6\cdot 31.62=429.72`

d. This value for x, that minimizes the average cost, happens when the first derivative of the average cost is equal to 0.

`c(x)=(C(x))/(x)=(54,000+240x+4x^(3/2))/(x)=54,000x^(-1)+240+4x^(1/2)\\\\\\ (dc)/(dx)=54,000(-1)x^(-2)+0+4(1/2)x^(-1/2)=0\\\\\\(dc)/(dx)=-54,000x^(-2)+2x^(-1/2)=0\\\\\\2x^(-1/2)=54,000x^(-2)\\\\\\x^(-1/2+2)=54,000/2=27,000\\\\\\x^(3/2)=27,000\\\\\\x=27,000^(2/3)=900`

e. The minimum average cost is:

`c(900)=54,000(900)^(-1)+240+4(900)^(1/2)\\\\c(900)=60+240+120\\\\c(900)=420`