Question

The Student Government Association is making Mother's Day gift baskets to sell at a fund-raiser. If the SGA makes a larger quantity of baskets, it can purchase materials in bulk. The total cost (in hundreds of dollars) of making x gift baskets can be approximated C(x) = 10x + 1/x + 100.

a. Find the marginal cost function and the marginal cost at x = 20 and x = 40.

b. Find the average-cost function and the average cost at x = 20 and x = 40.

c. Find the marginal average-cost function and the marginal average cost at x = 20 and x = 40.

Answer 1

a)

Marginal cost function = C'(x) = 999/(x+100)²

Marginal cost at x = 20 = C'(20) = $6.94

Marginal cost at x = 40 = C'(40) = $5.1

b)

Average cost function = A(x) = (10x + 1)/(x² + 100x)

Average cost at x = 20 = A(20) = $8.37

Average cost at x = 40 = A(40) = $7.16

c)

Marginal Average cost function = A'(x) = (10x² + 2x + 100)/(x² + 100x)²

Marginal Average cost at x = 20 = A'(20) = -$0.07

Marginal Average cost at x = 40 = A(40) = -$0.05

Explanation:

The cost function is given by

C(x) = (10x + 1)/(x + 100)

a. Find the marginal cost function and the marginal cost at x = 20 and x = 40

Taking the derivative of the cost function yields the marginal cost function.

Differentiate the cost function with respect to x

C'(x) = 10(x+100) - 1(10x +1)/(x+100)^2

C'(x) = (10x+1000 - 10x - 1)/(x+100)²

C'(x) = 999/(x+100)²

Evaluate the marginal cost function at x = 20 to get the marginal cost at x = 20

C'(20) = 999/(20+100)²

C'(20) = 999/14400

C'(20) = 0.0694

C'(20) = $6.94

Evaluate the marginal cost function at x = 40 to get the marginal cost at x = 40

C'(40) = 999/(40+100)²

C'(40) = 999/19600

C'(40) = 0.051

C'(40) = $5.1

b. Find the average-cost function and the average cost at x = 20 and x = 40

Dividing the cost function by x yields the average cost function.

A(x) = ((10x + 1)/(x + 100))/x

A(x) = (10x + 1)/(x² + 100x)

Evaluate the average cost function at x = 20 to get the average cost at x = 20

A(20) = (10*20 + 1)/(20² + 100*20)

A(20) = 201/2400

A(20) = 0.0837

A(20) = $8.37

Evaluate the average cost function at x = 40 to get the average cost at x = 40

A(40) = (10*40 + 1)/(40² + 100*40)

A(40) = 401/5600

A(40) = 0.0716

A(20) = $7.16

c. Find the marginal average-cost function and the marginal average cost at x = 20 and x = 40.

Taking the derivative of the average cost function yields the marginal average cost function.

A(x) = (10x + 1)/(x² + 100x)

A'(x) = (10x² + 1000x - 20x² - 1000x - 2x - 100)/(x² + 100x)²

A'(x) = (10x² + 2x + 100)/(x² + 100x)²

Evaluate the marginal average cost function at x = 20 to get the marginal average cost at x = 20

A'(20) = (10*20² + 2*20 + 100)/(20² + 100*20)²

A'(20) = -4140/2400²

A'(20) = -0.00072

A'(20) = -$0.07

Evaluate the marginal average cost function at x = 40 to get the marginal average cost at x = 40

A'(40) = (10*40² + 2*40 + 100)/(40² + 100*40)²

A'(40) = -16180/5600²

A'(40) = -0.00052

A'(40) = -$0.05