# Math

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
height:x+1
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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