Let:

x = width of box

h = height of box

V = Volume of box

A = Surface Area of box

P = Proportion of the height of the box to the side of the base

then:

length of box = 3x

Now for constant A we can make h a function of x by re-arranging formula for A as follows:

And so now we can calculate V as a function of x:

And to find the maximum V we need so:

But we also know that so:

So

Which is the solution:

Proportion of the height of the box to the side of the base is 1.5 !