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The Open Box Problem Equations for a square Total length of paper = x Total length of cutout = a Width of base of box = c = x - 2a Area of base = c2 Area of sides = 4ac Volume = ac2 Square piece of paper with dimensions 12 x 12. Cut out Length Width Area of base Area of sides Volume 1 10 10 100 40 100 2 8 8 64 64 128 2.1 7.8 7.8 60.84 65.52 127.764 2.2 7.6 7.6 57.76 66.88 127.072 2.3 7.4 7.4 54.76 68.08 125.948 2.4 7.2 7.2 51.84 69.12 124.416 2.5 7 7 49 70 122.5 2.6 6.8 6.8 46.24 70.72 120.224 2.7 6.6 6.6 43.56 71.28 117.612 2.8 6.4 6.4 40.96 71.68 114.688 2.9 6.2 6.2 38.44 71.92 111.476 3 6 6 36 72 108 3.1 5.8 5.8 33.64 71.92 104.284 3.2 5.6 5.6 31.36 71.68 100.352 3.3 5.4 5.4 29.16 71.28 96.228 3.4 5.2 5.2 27.04 70.72 91.936 3.5 5 5 25 70 87.5 3.6 4.8 4.8 23.04 69.12 82.944 3.7 4.6 4.6 21.16 68.08 78.292 3.8 4.4 4.4 19.36 66.88 73.568 3.9 4.2 4.2 17.64 65.52 68.796 4 4 4 16 64 64 5 2 2 4 40 20 A graph to show the comparison of the cutout to the volume of a 12 x 12 square. Square piece of paper with dimensions 16 x 16. Cut out Length Width Area of base Area of sides Volume 1 14 14 196 56 196 2 12 12 144 96 288 2.1 11.8 11.8 139.24 99.12 292.404 2.2 11.6 11.6 134.56 102.08 296.032 2.3 11.4 11.4 129.96 104.88 298.908 2.4 11.2 11.2 125.44 107.52 301.056 2.5 11 11 121 110 302.5 2.6 10.8 10.8 116.64 112.32 303.264 2.7 10.6 10.6 112.36 114.48 303.372 2.8 10.4 10.4 108.16 116.48 302.848 2.9 10.2 10.2 104.04 118.32 301.716 3 10 10 100 120 300 3.1 9.8 9.8 96.04 121.52 297.724 3.2 9.6 9.6 92.16 122.88 294.912 3.3 9.4 9.4 88.36 124.08 291.588 3.4 9.2 9.2 84.64 125.12 287.776 3.5 9 9 81 126 283.5 3.6 8.8 8.8 77.44 126.72 278.784 3.7 8.6 8.6 73.96 127.28 273.652 3.8 8.4 8.4 70.56 127.68 268.128 3.9 8.2 8.2 67.24 127.92 262.236 4 8 8 64 128 256 5 6 6 36 120 180 A graph to show the comparison of the cutout to the volume of a 16 x 16 square. In the first two charts the length of the cutout size has been 1/6 of the size of the original square. I have predicted that for a 20 x 20 square piece of paper that the maximum volume I can get will be with a cutout of 3cm. Square piece...

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