Set up and evaluate the optimization problem.

Question

You are constructing a cardboard box from a piece of cardboard with the dimensions 2 m by 4 m. You then cut equal-size squares from each corner so you may fold the edges. What are the dimensions (in m) of the box with the largest volume?

Answer 1

V(x) = x(2 - 2x)(4 - 2x)

Expand

Determine V'(x)

Set V'(x) = 0

Answer 2

Hello Azalea. So first you want to get your unknowns. The only unknown that we have is how big the squares we cut out are. We'll call the side length of the square "s".

Now we'll turn the cardboard into a box. This is a bit difficult to visualize. But if we cut out squares in the corner and then fold the top, bottom, left, and right flaps up, we get a box with its top missing. The box's dimensions are: height = s, width = 2 - 2s, and length = 4 - 2s.

Thus the volume is: s(2-2s)(4-2s)

Do you think you can find the maximum volume of the box now?