Folded boxes


a. Squares with sides of length x are cut out of each corner of a rectangular piece of cardboard measuring 5 ft by 8 ft. The resulting piece of cardboard is then folded into a box without a lid. Find the volume of the largest box that can be formed in this way.

Verified step by step guidance

1. Define the dimensions of the box after cutting out the squares: the length will be (8 - 2x) ft, the width will be (5 - 2x) ft, and the height will be x ft.
2. Write the volume V of the box as a function of x: V(x) = length * width * height = (8 - 2x)(5 - 2x)(x).
3. Expand the volume function V(x) to express it in a standard polynomial form.
4. Determine the domain of x, which is constrained by the dimensions of the cardboard: 0 < x < 2.5 ft.
5. Find the critical points of V(x) by taking the derivative V'(x), setting it to zero, and solving for x to identify the maximum volume.