Consider the following cost function. a. Find the average cost and marginal cost functions. b. Determine the average and marginal cost when xequalsa. c. Interpret the values obtained in part (b)

a) Average Cost function = 0.1 + (1000/x)

Marginal Cost function = 0.1

b) At x = a = 2000

Average Cost = 0.6

Marginal Cost = 0.1

c) The average cost calculate at x = 2000 in (b) represents the average cost of producing the first 2000 units of product and the marginal cost calculated at x = 2000 in (b) represents the cost of producing the 2001th unit of product.

Explanation:

The complete question

Consider the following cost functions.

a. Find the average cost and marginal cost functions.

b. Determine the average and marginal cost when x=a.

c. Interpret the values obtained in part (b).

C(x)=1000+0.1x, 0≤x≤5000, a=2000

Solution

a) The average cost is given as the total cost divided by the quantity produced.

A(x) = C(x) ÷ x

C(x) = 1000 + 0.1x

A(x) = (1000 + 0.1x) ÷ x = (1000/x) + 0.1

A(x) = 0.1 + (1000/x)

The marginal cost is given as the first derivative of the cost function with respect to the quantity of products produced.

M(x) = (dC/dx)

C(x) = 1000 + 0.1x

M(x) = (d/dx) (1000 + 0.1x) = 0.1

b) To calculate these values at x = a = 2000

Average cost at x = 2000

A(x) = 0.1 + (1000/x) = 0.1 + (1000/2000) = 0.1 + 0.5 = 0.6

Marginal Cost at x = 2000

M(x) = 0.1

c) The average cost is the cost per unit of producing a particular quantity of product.

The marginal cost is the cost of producing an extra unit of product.

Hence, the average cost calculate at x = 2000 in (b) represents the average cost of producing the first 2000 units of product and the marginal cost calculated at x = 2000 in (b) represents the cost of producing the 2001th unit of product.

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