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 $\displaystyle V$, you want to minimize (exterior) surface area.
Let $\displaystyle x=width$, $\displaystyle y=length$, and $\displaystyle z=height$
$\displaystyle V=xyz$
$\displaystyle S=xy+2xz+2yz$ (because there is no top)
You want to minimize $\displaystyle S$.
This looks like a good time to use Lagrange multipliers.
$\displaystyle \nabla S=\langle y+2z,x+2z,2x+2y\rangle$
$\displaystyle \nabla V=\langle yz,xz,xy\rangle$
($\displaystyle \nabla$ denotes the gradient.)
So you need to solve $\displaystyle \langle y+2z,x+2z,2x+2y\rangle=\lambda\langle yz,xz,xy\rangle$ given that $\displaystyle xyz=V$. (Remember that $\displaystyle V$ is a constant.)