# What will the dimensions of the resulting cardboard box be if the company wants to maximize the volume and they start with a flat piece of square cardboard 20 feet per side, and then cut smaller squares out of each corner and fold up the sides to create the box?

What will the dimensions of the resulting cardboard box be if the company wants to maximize the volume and they start with a flat piece of square cardboard 20 feet per side, and then cut smaller squares out of each corner and fold up the sides to create the box?
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coolng90qo
Suppose that the squares removed from each corner are x feet by x feet each.
When these are folded up they give a box with a height of x feetand a base of 20−2x feet by 20−2x feet for a volume
$V=x{\left(20-2x\right)}^{2}=400x-80{x}^{2}+4{x}^{3}$
To find the critical point(s) take the derivative of V, set it to zero, and solve for x.
$\frac{dV}{dx}=400-160x+12{x}^{2}$
$=4\left(3x-10\right)\left(x-10\right)=0$
Since x=10 gives a Volume of 0
so the critical point for the Volume that is it's maximum occurs when $x=\frac{10}{3}$
The resulting box will be
$3\frac{1}{3}×13\frac{1}{3}×13\frac{1}{3}$