Advanced Optimization: Calculus I and II (AB and BC)

Find the dimensions of the open-lid box with greatest volume that can be made from a rectangular sheet of paper measuring 11 by 14 inches, if the box is made by cutting congruent squares from the corners of the sheet and folding the resulting sides.

Difficulty:

Solution:

Look at the below diagram, which is festively festooned with holiday colors. On it, you see the rectangular paper in green with red squares to be cut out from each corner. Each square will have side x in length.

If x is the side of the square, then the sides of the resulting box will be (14 - 2x) and (11 - 2x), whereas the height of the box will be x. Since we are maximizing volume, use volume of a box as your primary formula:

Now, find the derivative of V and set it equal to 0 to maximize.

The second answer makes no sense, since 11 - 2x will produce a negative number, and we don't want our box to fold in on itself and rip space and time apart. Thus, the answer is x = 2.039, which you can verify with a wiggle graph. Thus, the dimensions of the box are x = 2.039 inches, (11 - 2x) = 6.922 inches, and (14 - 2x) = 9.922 inches.