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26 November, 03:47

You are under contract to design a storage building with a square base and a volume of 14,000 cubic feet. the cost of materials is $4 per square foot for the floor, $16 per square foot for the walls and $3 per square foot for the roof. find the dimensions that minimize the cost of materials.

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  1. 26 November, 05:53
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    The first thing we are going to do for this case is define variables.

    We have then:

    y = the cost of the box

    x = one side of the square base

    z = height of the box

    The volume of the building is 14,000 cubic feet:

    x ^ 2 * z = 14000

    We cleared z:

    z = (14000 / x ^ 2)

    On the other hand, the cost will be:

    floor = 4 (x ^ 2)

    roof = 3 (x ^ 2)

    for the walls:

    1 side = 16 (x * (14000 / x ^ 2)) = 16 (14000 / x)

    4 sides = 64 (14000 / x) = 896000 / x

    The total cost is:

    y = floor + roof + walls

    y = 4 (x ^ 2) + 3 (x ^ 2) + 896000 / x

    y = 7 (x ^ 2) + 896000 / x

    We derive the function:

    y ' = 14x - 896000 / x ^ 2

    We match zero:

    0 = 14x - 896000 / x ^ 2

    We clear x:

    14x = 896000 / x ^ 2

    x ^ 3 = 896000/14

    x = (896000/14) ^ (1/3)

    x = 40

    min cost (y) occurs when x = 40 ft

    Then,

    y = 7 * (40 ^ 2) + 896000/40

    y = 33600 $

    Then the height

    z = 14000/40 ^ 2 = 8.75 ft

    The price is:

    floor = 4 * (40 ^ 2) = 6400

    roof = 3 * (40 ^ 2) = 4800

    walls = 16 * 4 * (40 * 8.75) = 22400

    Total cost = $ 33600 (as calculated previously)

    Answer:

    The dimensions for minimum cost are:

    40 * 40 * 8.75
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