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The Open Box Problem

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Introduction

I had the task of making a box out of an A4 piece of paper with the largest possible volume. First of all I made a box out of an A4 piece of paper. To do this I had to cut squares out of each corner of the paper and fold up the sides. I used random measurements in this test to see the results. I found that the size of the squares I cut out of the corner changed the volume of the box I made, so, using a spreadsheet I tried to find the maximum volume. Box Made From A4 paper Corner Length Width Height Volume mms mms mms mms mm3 0 297 210 0 0 10 277 190 10 526300 20 257 170 20 873800 30 237 150 30 1066500 40 217 130 40 1128400 50 197 110 50 1083500 60 177 90 60 955800 70 157 70 70 769300 80 137 50 80 548000 90 117 30 90 315900 100 97 10 100 97000 I first found out the box volumes for box corner sizes 0 - 100mm. ...read more.

Middle

40.4 1128495 40.5 216 129 40.5 1128492 40.6 215.8 128.8 40.6 1128479 40.7 215.6 128.6 40.7 1128455 40.8 215.4 128.4 40.8 1128420 40.9 215.2 128.2 40.9 1128375 41 215 128 41 1128320 41.1 214.8 127.8 41.1 1128254 40.4mm produced the largest volume. The graph shows my results To produce more accurate results still, I used 2 decimal places, starting at 40.41mm, going up to 40.49mm. Box made from A4 paper Corner Length Width Height Volume mms mms mms mms mm3 40.41 216.18 129.18 40.41 1128495 40.42 216.16 129.16 40.42 1128495 40.43 216.14 129.14 40.43 1128495 40.44 216.12 129.12 40.44 1128495 40.45 216.1 129.1 40.45 1128495 40.46 216.08 129.08 40.46 1128494 40.47 216.06 129.06 40.47 1128494 40.48 216.04 129.04 40.48 1128493 40.49 216.02 129.02 40.49 1128493 Box made from A4 paper Corner Length Width Height Volume cms cms cms cms cm3 40.381 216.238 129.238 40.381 1128494 40.382 216.236 129.236 40.382 1128494 40.383 216.234 129.234 40.383 1128494 40.384 216.232 129.232 40.384 1128494 40.385 216.23 129.23 40.385 1128494 40.386 216.228 129.228 40.386 1128494 40.387 216.226 129.226 40.387 1128494 40.388 216.224 129.224 40.388 1128494 40.389 216.222 129.222 40.389 ...read more.

Conclusion

216.138 129.138 40.431 1128495 40.432 216.136 129.136 40.432 1128495 40.433 216.134 129.134 40.433 1128495 40.434 216.132 129.132 40.434 1128495 40.435 216.13 129.13 40.435 1128495 40.436 216.128 129.128 40.436 1128495 40.437 216.126 129.126 40.437 1128495 40.438 216.124 129.124 40.438 1128495 40.439 216.122 129.122 40.439 1128495 40.441 216.118 129.118 40.441 1128495 40.442 216.116 129.116 40.442 1128495 40.443 216.114 129.114 40.443 1128495 40.444 216.112 129.112 40.444 1128495 40.445 216.11 129.11 40.445 1128495 40.446 216.108 129.108 40.446 1128495 40.447 216.106 129.106 40.447 1128495 40.448 216.104 129.104 40.448 1128495 40.449 216.102 129.102 40.449 1128495 40.451 216.098 129.098 40.451 1128495 40.452 216.096 129.096 40.452 1128495 40.453 216.094 129.094 40.453 1128495 40.454 216.092 129.092 40.454 1128495 40.455 216.09 129.09 40.455 1128495 40.456 216.088 129.088 40.456 1128495 40.457 216.086 129.086 40.457 1128495 40.458 216.084 129.084 40.458 1128494 40.459 216.082 129.082 40.459 1128494 40.46 216.08 129.08 40.46 1128494 40.461 216.078 129.078 40.461 1128494 40.462 216.076 129.076 40.462 1128494 40.463 216.074 129.074 40.463 1128494 40.464 216.072 129.072 40.464 1128494 40.465 216.07 129.07 40.465 1128494 40.466 216.068 129.068 40.466 1128494 40.467 216.066 129.066 40.467 1128494 40.468 216.064 129.064 40.468 1128494 40.469 216.062 129.062 40.469 1128494 ...read more.

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