The dimensions of a gift box are consecutive positive integers such that the height is the least integer and the length is the greatest integer. If the height is increased by 1 cm, the width is increased by 2 cm, and the length is increased by 3 cm, then a larger box is constructed such that the volume is increased by 456 cm^3. Determine the dimensions of each box.

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  1. Original box
    height: x
    width : x+1
    length: x+2
    Volume = x(x+1)(x+2)

    new box
    width: x+3
    length: x+5
    volume = (x+1)(x+3)(x+5)

    so (x+1)(x+3)(x+5) - x(x+1)(x+2) = 456

    which when expanded, then simplified gave me the quadratic
    2x^2 + 7x - 147 = 0
    which factors to
    (x-7)(2x+21) = 0
    so x=7 or x = -21/2
    but x had to be an integer, so x=7

    sub back in to get the dimensions

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