A box can be created by removing squares from each corner of a rectangle of paper.

Let be the volume of the box in cubic inches, where is the side length, in inches, of each square removed from the four corners.

To define using an expression, we can use the fact that the volume of a cube is . If the piece of paper we start with is 3 inches by 8 inches, then:

What are some reasonable values for ? Cutting out squares with side lengths less than 0 inches doesn’t make sense, and similarly, we can’t cut out squares larger than 1.5 inches, since the short side of the paper is only 3 inches (since ). You may remember that the name for the set of all the input values that make sense to use with a function is the domain. Here, a reasonable domain is somewhere larger than 0 inches but less than 1.5 inches, depending on how well we can cut and fold!

By graphing this function, it is possible to find the maximum value within a specific domain. Here is a graph of  for values between 0 and 1.5. It looks like the largest volume we can get for a box made this way from a 3-inch by 8-inch piece of paper is about 7.4 in³.