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?
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
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.