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8 June, 13:51

A trash company is designing an open-top, rectangular container that will have a volume of 1715 ft cubed. The cost of making the bottom of the container is $5 per square foot, and the cost of the sides is $4 per square foot. Find the dimensions of the container that will minimize total cost.

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  1. 8 June, 14:03
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    14 ft * 14 ft * 8.75 ft

    Step-by-step explanation:

    A garbage company is designing an open rectangular container that should have a volume of 1,715 cubic feet.

    So we have the length of the container = "x" ft, the width of the container = "y" ft and the height of the container = "z" ft

    Therefore the volume of the rectangular container would be:

    x * y * z = 1715 ft³

    z = 1715 / x * y

    The cost of making the bottom of the container is $ 5 per square foot, that is:

    5 * (x * y)

    Now, area of all sides of the container would be:

    2 * (x * z + y * z) = 2 * z * (x + y)

    We know that it has been given that the cost of making all the sides of the container is = $ 4 per square foot, so:

    4 * (2 * z * (x + y)) = 8 * z * (x + y)

    In total the costs would be:

    5 * (x * y) + 8 * z * (x + y)

    If we replace z, in the previous equation we have:

    5 * (x * y) + 8 * (1715 / x * y) * (x + y)

    solving, and we would have that the total cost would be:

    C = 5 * (x * y) + 13720 / x + 13720 / y

    Now we will find the derivative of C and make it equal to zero:

    dC / dx = 0; dC / dy = 0

    For dC / dx = 0:

    dC / dx = 5 * y + 13720 * - 1 / (y ^ 2) + 13720 * 0

    0 = 5 * y - 13720 / y ^ 2

    5 * y = 13720 / y ^ 2

    y ^ 3 = 13720/5 = 2744

    y = 14

    For dC / dy = 0:

    dC / dy = 5 * x + 13720 * 0 + 13720 * - 1 / (x ^ 2)

    0 = 5 * x - 13720 / x ^ 2

    5 * x = 13720 / x ^ 2

    x ^ 3 = 13720/5 = 2744

    x = 14

    now for z:

    z = 1715 / (14 * 14)

    z = 8.75

    Therefore, the dimensions of the container should be 14 ft * 14 ft * 8.75 ft to minimize manufacturing cost.
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