Quote by mathwonk
well the thing it implies to me is that putting the two together should make a cube of area 123 sq ft, i.e. it should give the answer to the closed top problem. So the answer to the open top problem should be the result of cutting a cube of total area 12 sq ft in half, so the answer will have base sides twice its height. The base will then have side of length sqrt(2), and height half that. so the base will have area 2 and the 4 sides will each be rectangles of area 1.
what do you think? does that make sense? try it using calculus and see what happens.

yes that does make sense. using thought experiments makes it a very out of the box (pun intended) solution. but analytically, it is very apparent.