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.

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

Step-by-step 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