The sales department at OHaganBooks.com, which has decided to send chocolate lobsters to each of its best customers, is trying to design a shipping box with a square base. It has a roll of cardboard 42 inches wide from which to make the boxes. Each box will be obtained by cutting out corners from a rectangle of cardboard as shown in the following diagram. (Notice that the top and bottom of each box will be square, but the sides will not necessarily be square.)

42"

A cutout of a box has six pieces.

The first piece is square and is labeled "bottom."

Along the left, right, top, and bottom sides of the "bottom" piece is a rectangular piece labeled "side." The longer edge of each "side" piece is attached to an edge of the "bottom" piece.

Along the right side of the "side" piece that is attached to the right side of the "bottom" piece is a square piece labeled "top." It it the same size as the "bottom" piece.

The length from the left side of the cutout to the right side is 42 inches.

What are the dimensions of the boxes with the largest volume that can be made in this way?

in (smallest value) in in (largest value)

What is the maximum volume?