It works out the the optimum box dimensions are those that make the lateral area twice the base area. Of course, the base is a square in order to minimize the lateral area for a given base area.
That being the case, the height of the box is half the base edge length and the dimensions are
.. x = y = 6 in
.. z = 3 in
_____
If x and y are the base dimensions, the volume is
.. xyz = 108
and we want to minimize the area
.. area = xy +2(x+y)z
These equations are symmetrical in x and y, so the solution will require x=y. Making that substitution, we have
.. x^2*z = 108
.. x^2 +4xz = area = x^2 +4x(108/x^2)
That is, we want to minimize x^2 +432/x. A graphing calculator finds the minimum at x=y=6, (and z = 108/x^2 = 3).
