A box with a square base and open top must have a volume of 442368 cm3. We wish to find the dimensions of the box that minimize the amount of material used.

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'(x).

A'(x)=

Now, calculate when the derivative equals zero, that is, when A'(x)=0. [Hint: multiply both sides by x2.]

A'(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"(x).

A"(x)=

Evaluate A"(x) at the x-value you gave above.