For the second part, the only value that changes is the length. Instead of 11 inches, the length is now 14 inches. The process of computing the volume is the same, but the formula for length changes. Instead of our formula being 11-2x, the formula is now 14-2x.

I create a separate column for my new initial length, but I leave the width and height untouched. I begin the column with 14, and I continue down the column filling each cell with the appropriate value given the size of square cuts.

Finally, I compute the volume with my new length values. This is the same process as before, only the length value changes. Remember, the volume is the product of length, width, and height. Using the cell actions in the spreadsheet, I multiply to find the volume.

I analyze the different volume values produced for each value of “x”. I notice that the largest volume (approximately 92 inches squared) occurs when the length is 10.5 inches, the width is 5 inches, and the height is 1.75 inches. You can also explore different increments with the “x” value, but the volume does surpass 92 inches squared.

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The last part of the question asks for a generalization of a rectangular piece of paper with any dimension. For a rectangular piece of paper of any dimension, the height is going to play a key role in determining the volume, and the height is dependent on the length and width of the paper. If you have a smaller piece of rectangular paper, you will have little to work with when building the height of your box, and therefore, a smaller volume. The larger your length and width, the more you can use for your height. Therefore, your volume will be larger.

Now, it is time to discuss the SMPs and Alabama College and Career Ready Standards that are found in this problem.

Or, go back to the first page.