1. ## Calc III

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.

2. Originally Posted by tdat1979
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 $V$, you want to minimize (exterior) surface area.

Let $x=width$, $y=length$, and $z=height$

$V=xyz$
$S=xy+2xz+2yz$ (because there is no top)

You want to minimize $S$.

This looks like a good time to use Lagrange multipliers.

$\nabla S=\langle y+2z,x+2z,2x+2y\rangle$
$\nabla V=\langle yz,xz,xy\rangle$

( $\nabla$ denotes the gradient.)

So you need to solve $\langle y+2z,x+2z,2x+2y\rangle=\lambda\langle yz,xz,xy\rangle$ given that $xyz=V$. (Remember that $V$ is a constant.)