A box with a square base and open top must have a volume of $442368c{m}^3.$ We wish to find the dimensions of the box that minimize the amount of material used. $[8$ points$]$

Follow the following steps:

First, find a formula for the surface area of the box in terms of only $x,$ the length of one side of the square base.

$[$Hint: use the volume formula to express the height of the box in terms of $x.]$ Simplify your formula as much as possible.

$A(x)=$

Next, find the derivative, $A^{\text{'}}(x).$

$2.A^{\text{'}}(x)=$

Now, calculate when the derivative equals zero, that is$,$ when $A^{\text{'}}(x)=0.[$Hint: multiply both sides by $x^2.]$

$3.A^{\text{'}}(x)=0$ when $x=$

We next have to make sure that this value of $x$ gives a minimum value for the surface area. Let's use the second derivative test. Find $A^{\text{'}\text{'}}(x).$

$4.A^{\text{'}\text{'}}(x)=$

Evaluate $A^{\text{'}\text{'}}(x)$ at the $x-$value you gave above.