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8 September, 08:13

A closed box with a square base is to have a volume of 85. 75 cm^3. the material for the top and bottom of the box costs $3.00 per square centimeter, while the material for the sides costs $1.50 per square centimeter. find the dimensions of the box that will lead to the minimum total cost. what is the minimum total cost?

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  1. 8 September, 08:40
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    We can create two equations here:

    (1) Volume = area of square * height of box

    85.75 = s^2 h

    (2) Cost = 3 * area of square + 1.5 * area of side box

    C = 3 s^2 + 1.5 s h

    From (1), we get:

    h = 85.75 / s^2

    Combining this with (2):

    C = 3 s^2 + 1.5 s (85.75 / s^2)

    C = 3 s^2 + 128.625 s-

    Taking the 1st derivative and equating dC/ds = 0:

    dC/ds = 6s - 128.625 / s^2 = 0

    Multiply all by s^2:

    6s^3 - 128.625 = 0

    6s^3 = 128.625

    s = 2.78 cm

    So h is:

    h = 85.75 / s^2 = 85.75 / (2.78) ^2

    h = 11.10 cm

    So the dimensions are 2.78 cm x 2.78 cm x 11.10 cm

    The total cost now is:

    C = 3 (2.78) ^2 + 1.5 (2.78) (11.10)

    C = $69.47
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