A box with a square base and open top must have a volume of 70304 $\displaystyle {c}{m}^{{3}}$. We wish to find the dimensions of the box that minimize the amount of material used.

First, find a formula for the (external) surface area of the box in terms of only $\displaystyle {x}$, the length of one side of the square base.
[Hint: use the volume formula to express the height of the box in terms of $\displaystyle {x}$.]
Simplify your formula as much as possible.
$\displaystyle {A}{\left({x}\right)}=$Preview Question 6 Part 1 of 4  

Next, find the derivative, $\displaystyle {A}'{\left({x}\right)}$.
$\displaystyle {A}'{\left({x}\right)}=$Preview Question 6 Part 2 of 4  

Now, calculate when the derivative equals zero, that is, when $\displaystyle {A}'{\left({x}\right)}={0}$.
$\displaystyle {A}'{\left({x}\right)}={0}$ when $\displaystyle {x}=$

Verify by looking at a graph that this is indeed the $\displaystyle {x}$-value that minimizes the function $\displaystyle {A}{\left({x}\right)}$. Then, using the value of $\displaystyle {x}$ you just found, find the surface area of the box.
Area=`