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16 March, 15:34

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.

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  1. 16 March, 16:04
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    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
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