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3 April, 22:33

An open box is constructed of 3500 cm2 of cardboard. The box is a cuboid, with height hcm and square base of side xcm. What is the value of x which maximises the volume of the box?

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  1. 4 April, 02:23
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    34.16 cm

    Step-by-step explanation:

    side of square base = x

    height = h

    area, A = 3500 cm^2

    Area = x² + 4xh = 3500

    4 xh = 3500 - x²

    h = (3500 - x²) / 4x

    Volume = Area of base x height

    V = x² h

    V = x² (3500 - x²) / 4x

    V = (3500 x - x³) / 4

    Differentiate volume with respect to x

    dV/dx = (3500 - 3x²) / 4

    It is equal to zero for maxima and minima

    3500 - 3x² = 0

    x = 34.16 cm

    Now differentiate again

    d²V/dx² = 6x / 4

    It is negative so the volume is maximum.

    Thus, for x = 34.16 cm, the volume is maximum.
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