Answer
(a) $z^2+\frac{48}{z}$
(b) height $1.443ft$, base width $2.884ft$
Work Step by Step
(a) We assume the height is $x$ and side length is $z$ for the square base.
The volume is $xz^2=12$ which gives $x=\frac{12}{z^2}$
The surface area $A$ can be found as $A=z^2+4xz=z^2+4z\times\frac{12}{z^2}=z^2+\frac{48}{z}$
(b) Graph the above function as shown in the figure. A minimum in the surface area can be found at
$A=24.961ft^2$ when $z\approx2.884ft$ which gives $x\approx1.443ft$. Thus, to minimize the amount of material used, the height of the box should be about $1.443ft$ and the base width should be about $2.884ft$
