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• Level: GCSE
• Subject: Maths
• Word count: 1845

# The Open Box Problem

Extracts from this document...

Introduction

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

Middle

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 of paper with dimensions 20 x 20

 Cut out Length Width Area of base Area of sides Volume 1 18 18 324 72 324 2 16 16 256 128 512 2.1 15.8 15.8 249.64 132.72 524.244 2.2 15.6 15.6 243.36 137.28 535.392 2.3 15.4 15.4 237.16 141.68 545.468 2.4 15.2 15.2 231.04 145.92 554.496 2.5 15 15 225 150 562.5 2.6 14.8 14.8 219.04 153.92 569.504 2.7 14.6 14.6 213.16 157.68 575.532 2.8 14.4 14.4 207.36 161.28 580.608 2.9 14.2 14.2 201.64 164.72 584.756 3 14 14 196 168 588 3.1 13.8 13.8 190.44 171.12 590.364 3.2 13.6 13.6 184.96 174.08 591.872 3.3 13.4 13.4 179.56 176.88 592.548 3.4 13.2 13.2 174.24 179.52 592.416 3.5 13 13 169 182 591.5 3.6 12.8 12.8 163.84 184.32 589.824 3.7 12.6 12.6 158.76 186.48 587.412 3.8 12.4 12.4 153.76 188.48 584.288 3.9 12.2 12.2 148.84 190.32 580.476 4 12 12 144 192 576 5 10 10 100 200 500

A graph to show the comparison of the cutout to the volume of a 19 x 19 square.

The formula for the maximum volume of the box for square shaped pieces of paper is:

x = side of square

/ = divide by

x/6 = side of square / 6

Conclusion

="c1">56

108

168

4

12

2

24

112

96

A graph to show the comparison of the cutout to the volume of a 10 x 20 rectangle.

Rectangle piece of paper with dimensions 10 x 30.

 Cut out Width Length Area of base Area of sides Volume 1 28 8 224 72 224 1.1 27.8 7.8 216.84 78.32 238.524 1.2 27.6 7.6 209.76 84.48 251.712 1.3 27.4 7.4 202.76 90.48 263.588 1.4 27.2 7.2 195.84 96.32 274.176 1.5 27 7 189 102 283.5 1.6 26.8 6.8 182.24 107.52 291.584 1.7 26.6 6.6 175.56 112.88 298.452 1.8 26.4 6.4 168.96 118.08 304.128 1.9 26.2 6.2 162.44 123.12 308.636 2 26 6 156 128 312 2.1 25.8 5.8 149.64 132.72 314.244 2.2 25.6 5.6 143.36 137.28 315.392 2.3 25.4 5.4 137.16 141.68 315.468 2.4 25.2 5.2 131.04 145.92 314.496 2.5 25 5 125 150 312.5 2.6 24.8 4.8 119.04 153.92 309.504 2.7 24.6 4.6 113.16 157.68 305.532 2.8 24.4 4.4 107.36 161.28 300.608 2.9 24.2 4.2 101.64 164.72 294.756 3 24 4 96 168 288 4 22 2 44 192 176

A graph to show the comparison of the cutout to the volume of a 10 x 30 rectangle.

Rectangle piece of paper with dimensions 10 x 40.

 Cut out Width Length Area of base Area of sides Volume 1 38 8 304 92 304 1.1 37.8 7.8 294.84 100.32 324.32 1.2 37.6 7.6 285.76 108.48 342.91 1.3 37.4 7.4 276.76 116.48 359.79 1.4 37.2 7.2 267.84 124.32 374.98 1.5 37 7 259 132 388.5 1.6 36.8 6.8 250.24 139.52 400.38 1.7 36.6 6.6 241.56 146.88 410.65 1.8 36.4 6.4 232.96 154.08 419.33 1.9 36.2 6.2 224.44 161.12 426.44 2 36 6 216 168 432 2.1 35.8 5.8 207.64 174.72 436.04 2.2 35.6 5.6 199.36 181.28 438.59 2.3 35.4 5.4 191.16 187.68 439.67 2.4 35.2 5.2 183.04 193.92 439.3 2.5 35 5 175 200 437.5 2.6 34.8 4.8 167.04 205.92 434.3 2.7 34.6 4.6 159.16 211.68 429.73 2.8 34.4 4.4 151.36 217.28 423.81 2.9 34.2 4.2 143.64 222.72 416.56 3 34 4 136 228 408 4 32 2 64 272 256

A graph to show the comparison of the cutout to the volume of a 10 x 40 rectangle.

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