I wanted to begin by making sense of the problem and determining what was being asked of me. My first instinct was to get out a piece of paper and follow the steps of cutting and folding. This allowed me to physically see the model, rather than trying to imagine it in my head. I explored the effects of the box size based on the change in height. The height depended on the size of square cuts made in the corners of the paper. The larger the square cuts, the higher the sides of the box became.

(The values depicted are not related to our presented problem. The image is for visualization purposes only.)

Consequently, the larger square cuts decreased the length and width dimensions. Thus, if you begin with length of 11 inches, your formula for calculating the length, considering your square cuts, would be 11-2x. This formula is correct because your initial length is 11, but you will be taking away an “x” amount from both ends of the length of paper, hence why “x” is multiplied by 2.

The same situation occurs when calculating the width. Your initial width is 8.5 inches, but you will subtracting the “x” amount from both ends of the width. Again, like with the length formula, you will take your initial value and subtract 2x. Therefore, the formula for the width, considering the square cuts, would be 8.5-2x.

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Part One

Now, for the first part, I am asked to determine the dimensions that produce the largest box. I decided using a spreadsheet would be tremendously beneficial because it saves time and is organized. The app I used is called Numbers.

I begin by having my first column represent the square cuts that will be made at each corner of the paper. I do not know an exact value for the square cuts, but I know that they cannot be greater than or equal to 4.25 inches. If they are, there will not be enough paper for the box to have a width.

Think about this, if the “x” value is 4.25 inches, then 8.5-2(4.25) is zero. The “x” value will work for the length since the length is 11 inches, and 11-2(4.25) is 2.5 inches.

Since I know that “x” cannot be greater than or equal to 4.25 inches, I start my column with zero. Then I increase the “x” value by .25 inches. I chose to increase by .25 inches because I wanted to make sure I checked multiple cases. When I added one inch each time, I felt like I did not check enough possible “x” values to confidently determine the dimensions needed for the largest box.

I fill out the “x” column until I reach 4 inches, since I know “x” cannot equal 4.25 inches.

Next, I create a length column, I begin the column with 11 inches since that is our initial value. I already determined the formula for computing the length given the square cuts, so I use the cell actions in the spreadsheet to enter my formula.

Now, I complete the same process as above, but using the width formula. I begin the column with my initial value of 8.5 inches. Using the cell actions in the spreadsheet, I enter the formula determined earlier for width.

I mentioned earlier that the height depended on the size of the square cuts made in the corner of the paper. So, for the height column, I use the same values I listed for “x”. To find the volume of the box, we multiply length by width by height.

Finally, for the volume column, I use the cell actions in the spreadsheet to multiply length by width by height for each value of “x”. Once I have those volume values, I see that the largest volume (approximately 66 inches squared) occurs when the length is 8 inches, the width is 5.5 inches, and the height is 1.5 inches.

You could explore how the length and width values change when the values of “x” increase by more than .25 inches or when the values increase by less than .25 inches. You will notice that, even with different values, the volume is never an inch over 66 inches squared.

Now, on to part two and three of the problem.

Or, go back to the first page.