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27 January, 14:59

A box with an open top has vertical sides, a square bottom, and a volume of 108 cubic meters. if the box has the least possible surface area, find its dimensions. (in your answer leave a space between the number and the unit.)

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  1. 27 January, 18:25
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    Let x be the height of the box and y be the length of one side of the base then:-

    V = xy^2 = 108

    x = 108/y^2

    Surface area = y^2 + 4xy

    S = y^2 + 4y * 108/y^2

    S = y^2 + 432/y

    Finding the derivative:-

    dS/dy = 2y - 432/y^2 = 0

    2y^3 = 432

    y^3 = 216

    y = 6

    Check if this gives a minimum value:-

    second derivative = 2 + 864/y^3 which is positive so minimum.

    V = xy^2 = 108

    36y = 108

    y = 3

    Answer : - dimensions of the box is 3*6*6 metres
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