Determine the dimensions of a rectangular box, without a top, having volume V ft³, which requires the least amount of material for its construction.
I have no idea where to begin.
Conceptually, given a fixed volume , you want to minimize (exterior) surface area.
Let , , and
(because there is no top)
You want to minimize .
This looks like a good time to use Lagrange multipliers.
( denotes the gradient.)
So you need to solve given that . (Remember that is a constant.)