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25 June, 20:24

A rectangular tank with a square base, an open top, and a volume of 19 comma 65219,652 ft cubedft3 is to be constructed of sheet steel. Find the dimensions of the tank that has the minimum surface area.

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  1. 26 June, 00:20
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    34 and 17 feet.

    Step-by-step explanation:

    We have that the volume is given by:

    V = x * x * y = (x ^ 2) * y

    The surface area of the tank would be: x ^ 2 + x * y + x * y + x * y + x * y

    SA = x ^ 2 + 4 * x * y

    we know that the volume is 19652, replacing it would remain:

    19652 = (x ^ 2) * y, we solve for y:

    y = 19652 / (x ^ 2)

    replacing in SA we are left with:

    SA = x ^ 2 + 4 * x * 19652 / (x ^ 2)

    SA = x ^ 2 + 70608 / x

    we derive and equal 0:

    SA' (x) = 2 * x - 70608 / (x ^ 2) = 0

    2 * x = 70608 / (x ^ 2)

    x ^ 3 = 39304

    x = 34

    for and would be:

    y = 19652 / (34 ^ 2) = 17

    dimensions are 34 and 17 feet.
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