Answer
$S(w)=w^2+\frac{8}{w}$
$w>0$
Work Step by Step
Formulas used
volume $v=lwh$
area $a=lw$
total surface area = base area + 4(side area)
We are told the volume of the box is $2m^3$, so $v=2$.
We are also told the base is square. So we know the $4$ sides of the box will have equal areas.
Fill in the volume formula with what we know
$2=w*w*h$
Multiply
$2=w^2h$
Divide
$\frac{2}{w^{2}}=\frac{w^{2}h}{w^{2}}$
Simplify
$\frac{2}{w^{2}}=h$
Find the area of the base
$a=w^2$
Find the area of a side using the height we found above
$a=wh$
$a=w\left(\frac{2}{w^{2}}\right)$
$a=\frac{2w}{w^{2}}$
$a=\frac{2}{w}$
Find the total surface area
total surface area = base area + $4$(side area)
$S=w^2+4\left(\frac{2}{w}\right)$
$S=w^2+\frac{8}{w}$
Re-write as a function
$S(w)=w^2+\frac{8}{w}$
Since length can't be negative, and denominators can't be zero, we know $w$ must be greater than zero:
$w>0$
