# A box with a square base and open top must have a volume of 32,000cm^3. How do you find the dimensions of the box that minimize the amount of material used?

##### 1 Answer

The Volume of a box with a square base

The amount of material used is directly proportional to the surface area, so we will minimize the amount of material by minimizing the surface area.

The surface area of the box described is

We need

which gives us

We want to minimize

Which occurs when

The only critical number is

The second derivative test verifies that

**The box should have base 40 cm by 40 cm and height 20 cm.**

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