Answer
$4\times4\times3$ ft, $288$ dollars.
Work Step by Step
Step 1. Assume the edge length of the base square is $x$ ft and the height is $h$ ft. The volume is given by $V=x^2h=48$ and $h=\frac{48}{x^2}$
Step 2. The total cost can be expressed as $C=6x^2+4(4xh)=6x^2+16x(\frac{48}{x^2})=6x^2+\frac{768}{x}$
Step 3. To find the minimum cost, let its derivative to be zero; we have $C'=12x-\frac{768}{x^2}=0$, which gives $x=4$ ft, $h=3$ ft, and $C(4)=6(4)^2+\frac{768}{4}=288$ dollars.
Step 4. Check $C''=12+\frac{1536}{x^2}\gt0$ and we know the region is concave up with a minimum.
