A 12cm by 8cm rectangular piece of metal is to be made into an open-top box by cutting a sqaure from corner and folding up the resulting flaps (sides). If the volume of the lidless box is 36 cm what are the integer dimensions of the box?

Answer 1

Square to be cut out : x cm by x cm
then the height of the box is x, and the base is
12-2x by 8-2x, where 0 < x < 4

volume = x(12-2x)(8-2x)
= 36
x(6-x)(4-x) = 9
x(24 - 10x + x^2) = 9
x^3 - 10x^2 + 24x - 9 = 0
I tried x = ±1, ±3 and x = 3 was a solution.
by synthetic division, I got
(x-3)(x^2 - 7x + 3) = 0
the quadratic has no real solutions, so

x = 3 and the box will be
6 by 2 by 3

Answer 2

hi,

how did you go from 36=x(12-2x)(8-2x) to 9=x(6-x)(4-x)?