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• Level: GCSE
• Subject: Maths
• Word count: 6215

# The Open Box Problem

Extracts from this document...

Introduction

The Open Box

An open box is to be made from a sheet of card. Identical squares are cut off the Four Corners of the card as shown.

10

The card is then folded along the dotted lines to make the box. The main aim of this activity is to determine the size of the square cut which makes the volume of the box as large as possible for any given rectangular sheet of card.

1. For any sized square sheet of card investigate the size of the cut out square corner which makes an open box of the largest volume.
1. For any sized rectangular sheet of card investigate size of the cut out square corner which makes an open box of the largest volume.

We will use a ten-cm square. We are using a square because it is easier to use one variable because a squares width and length are the same.

 Cut size(cm) Original length Original width Width Length Height Volume 1 10 10 8 8 1 64 2 10 10 6 6 2 72 3 10 10 4 4 3 48 4 10 10 2 2 4 16

Therefore we can see that the maximum box area is made from the cut size of 2cms. Now I will try between 1-2 cm’s.

 Cut size Original length Original width Width Length Height Volume 1.1 10 10 7.8 7.8 1.1 66.924 1.2 10 10 7.6 7.6 1.2 69.312 1.3 10 10 7.4 7.4 1.3 71.188 1.4 10 10 7.2 7.2 1.4 72.576 1.5 10 10 7 7 1.5 73.5 1.6 10 10 6.8 6.8 1.6 73.984 1.7 10 10 6.6 6.6 1.7 74.052 1.8 10 10 6.4 6.4 1.8 73.728 1.9 10 10 6.2 6.2 1.9 73.036 2 10 10 6 6 2 72

The highest volume is 1.7, now i will try the maximum volumes between 1.6 and 1.7.

 Cut size Original length Original width Width Length Height Volume 1.61 10 10 6.78 6.78 1.61 74.0091 1.62 10 10 6.76 6.76 1.62 74.0301 1.63 10 10 6.74 6.74 1.63 74.047 1.64 10 10 6.72 6.72 1.64 74.0598 1.65 10 10 6.7 6.7 1.65 74.0685 1.66 10 10 6.68 6.68 1.66 74.0732 1.67 10 10 6.66 6.66 1.67 74.0739 1.68 10 10 6.64 6.64 1.68 74.0705 1.69 10 10 6.62 6.62 1.69 74.0632 1.7 10 10 6.6 6.6 1.7 74.052

Middle

15.7768

5.7768

2.5

192.45

2.1117

20

10

15.7766

5.7766

2.6

192.45

2.1118

20

10

15.7764

5.7764

2.7

192.45

2.1119

20

10

15.7762

5.7762

2.8

192.45

2.112

20

10

15.776

5.776

2.9

192.45

2.1121

20

10

15.7758

5.7758

3

192.45

As you can tell the volume does not change between 2.1113 and 2.1121. Therefore I shall not go any smaller. At the moment I cannot see anything, which may form a pattern so I will try another Rectangle.

Now I shall try a 40cm by 20cm rectangle.

 Cut Size Original Length Original Width Length Width Height Volume 1 40 20 38 18 2 684 2 40 20 36 16 2.1 1152 3 40 20 34 14 2.2 1428 4 40 20 32 12 2.3 1536 5 40 20 30 10 2.4 1500 6 40 20 28 8 2.5 1344 7 40 20 26 6 2.6 1092 8 40 20 24 4 2.7 768 9 40 20 22 2 2.8 396 10 40 20 20 0 2.9 0

The maximum volume is between 4cm and 5cm.

Now I will try from 4.0 to 5.0.

 Cut Size Original Length Original Width Length Width Height Volume 4 40 20 32 12 4 1536 4.1 40 20 31.8 11.8 4.1 1538.484 4.2 40 20 31.6 11.6 4.2 1539.552 4.3 40 20 31.4 11.4 4.3 1539.228 4.4 40 20 31.2 11.2 4.4 1537.536 4.5 40 20 31 11 4.5 1534.5 4.6 40 20 30.8 10.8 4.6 1530.144 4.7 40 20 30.6 10.6 4.7 1524.492 4.8 40 20 30.4 10.4 4.8 1517.568 4.9 40 20 30.2 10.2 4.9 1509.396 5 40 20 30 10 5 1500

This shows that the maximum Volume is at 1540 cm3.

Now I shall try between 4.2and 4.4.

 Cut Size Original Length Original Width Length Width Height Volume 4.1 40 20 31.8 11.8 4.1 1538.484 4.11 40 20 31.78 11.78 4.11 1538.654 4.12 40 20 31.76 11.76 4.12 1538.81 4.13 40 20 31.74 11.74 4.13 1538.952 4.14 40 20 31.72 11.72 4.14 1539.08 4.15 40 20 31.7 11.7 4.15 1539.194 4.16 40 20 31.68 11.68 4.16 1539.293 4.17 40 20 31.66 11.66 4.17 1539.379 4.18 40 20 31.64 11.64 4.18 1539.451 4.19 40 20 31.62 11.62 4.19 1539.508 4.2 40 20 31.6 11.6 4.2 1539.552 4.21 40 20 31.58 11.58 4.21 1539.582 4.22 40 20 31.56 11.56 4.22 1539.598 4.23 40 20 31.54 11.54 4.23 1539.6 4.24 40 20 31.52 11.52 4.24 1539.588 4.25 40 20 31.5 11.5 4.25 1539.563 4.26 40 20 31.48 11.48 4.26 1539.523 4.27 40 20 31.46 11.46 4.27 1539.47 4.28 40 20 31.44 11.44 4.28 1539.403 4.29 40 20 31.42 11.42 4.29 1539.322 4.3 40 20 31.4 11.4 4.3 1539.228

This shows where the Volume reaches its maximum.

I still cannot see a pattern so I shall try a 50 by 25 rectangle.

 Cut Size Original Length Original Width Length Width Height Volume 1 50 25 48 23 1 1104 2 50 25 46 21 2 1932 3 50 25 44 19 3 2508 4 50 25 42 17 4 2856 5 50 25 40 15 5 3000 6 50 25 38 13 6 2964 7 50 25 36 11 7 2772 8 50 25 34 9 8 2448 9 50 25 32 7 9 2016 10 50 25 30 5 10 1500 11 50 25 28 3 11 924 12 50 25 26 1 12 312

As we can see the maximum volume is between 5 & 6.

Now I will go up in 1 decimal place.

 Cut Size Original Length Original Width Length Width Height Volume 5 50 25 40 15 5 3000 5.1 50 25 39.8 14.8 5.1 3004.104 5.2 50 25 39.6 14.6 5.2 3006.432 5.3 50 25 39.4 14.4 5.3 3007.008 5.4 50 25 39.2 14.2 5.4 3005.856 5.5 50 25 39 14 5.5 3003 5.6 50 25 38.8 13.8 5.6 2998.464 5.7 50 25 38.6 13.6 5.7 2992.272 5.8 50 25 38.4 13.4 5.8 2984.448 5.9 50 25 38.2 13.2 5.9 2975.016 6 50 25 38 13 6 2964

This shows that the Maximum volume is between 3000 and 3020. Now I will try between 5.2 and 5.3 with a difference of two decimal places.

 Cut Size Original Length Original Width Length Width Height Volume 5.2 50 25 39.6 14.6 5.2 3006.432 5.21 50 25 39.58 14.58 5.21 3006.568 5.22 50 25 39.56 14.56 5.22 3006.687 5.23 50 25 39.54 14.54 5.23 3006.788 5.24 50 25 39.52 14.52 5.24 3006.871 5.25 50 25 39.5 14.5 5.25 3006.938 5.26 50 25 39.48 14.48 5.26 3006.986 5.27 50 25 39.46 14.46 5.27 3007.018 5.28 50 25 39.44 14.44 5.28 3007.032 5.29 50 25 39.42 14.42 5.29 3007.029 5.3 50 25 39.4 14.4 5.3 3007.008

Now I will try between 5.28 and 5.3 to try and find the maximum volume. I will use three decimal places.

Cut Size

Original Length

Original Width

Length

Width

Height

Volume

5.28

50

25

39.44

14.44

5.28

3007.032

5.281

50

25

39.438

14.438

5.281

3007.032

5.282

50

25

39.436

14.436

5.282

3007.033

5.283

50

25

39.434

14.434

5.283

3007.033

5.284

50

25

39.432

14.432

5.284

3007.033

5.285

50

25

39.43

14.43

5.285

3007.032

5.286

50

25

39.428

14.428

5.286

3007.032

5.287

50

25

39.426

14.426

5.287

3007.031

5.288

Conclusion

 Cut Size Original Length Original Width Length Width Height Volume 1 100 10 98 8 1 784 2 100 10 96 6 2 1152 3 100 10 94 4 3 1128 4 100 10 92 2 4 736 5 100 10 90 0 5 0

From this data we can tell that the maximum volume is between 2 and 3. Now I shall use one decimal place to see which cut size will give a larger volume.

 Cut Size Original Length Original Width Length Width Height Volume 2 100 10 96 6 2 1152 2.1 100 10 95.8 5.8 2.1 1166.844 2.2 100 10 95.6 5.6 2.2 1177.792 2.3 100 10 95.4 5.4 2.3 1184.868 2.4 100 10 95.2 5.2 2.4 1188.096 2.5 100 10 95 5 2.5 1187.5 2.6 100 10 94.8 4.8 2.6 1183.104 2.7 100 10 94.6 4.6 2.7 1174.932 2.8 100 10 94.4 4.4 2.8 1163.008 2.9 100 10 94.2 4.2 2.9 1147.356 3 100 10 94 4 3 1128

From this new data we can tell that the maximum volume is between 2.4 and 2.5. Now I shall use two decimal places to see what cut size will give the largest volume.

 Cut Size Original Length Original Width Length Width Height Volume 2.4 100 10 95.2 5.2 2.4 1188.096 2.41 100 10 95.18 5.18 2.41 1188.208 2.42 100 10 95.16 5.16 2.42 1188.282 2.43 100 10 95.14 5.14 2.43 1188.318 2.44 100 10 95.12 5.12 2.44 1188.315 2.45 100 10 95.1 5.1 2.45 1188.275 2.46 100 10 95.08 5.08 2.46 1188.196 2.47 100 10 95.06 5.06 2.47 1188.079 2.48 100 10 95.04 5.04 2.48 1187.924 2.49 100 10 95.02 5.02 2.49 1187.731 2.5 100 10 95 5 2.5 1187.5

The largest volume is made from a cut size of 2.43. Now I shall try 3 decimal places between 2.43 and 2.44.

 Cut Size Original Length Original Width Length Width Height Volume 2.43 100 10 95.14 5.14 2.43 1188.318 2.431 100 10 95.138 5.138 2.431 1188.319 2.432 100 10 95.136 5.136 2.432 1188.32 2.433 100 10 95.134 5.134 2.433 1188.321 2.434 100 10 95.132 5.132 2.434 1188.321 2.435 100 10 95.13 5.13 2.435 1188.321 2.436 100 10 95.128 5.128 2.436 1188.321 2.437 100 10 95.126 5.126 2.437 1188.32 2.438 100 10 95.124 5.124 2.438 1188.319 2.439 100 10 95.122 5.122 2.439 1188.317 2.44 100 10 95.12 5.12 2.44 1188.315

As we can tell between 2.433 and 2.436 the volume is the same.

Now I shall draw a graph to show my results-

Now I shall try between 2.433 and 2.436 to see if I can get a more accurate result.

 Cut Size Original Length Original Width Length Width Height Volume 2.433 100 10 95.134 5.134 2.433 1188.321 2.4331 100 10 95.1338 5.1338 2.4331 1188.321 2.4332 100 10 95.1336 5.1336 2.4332 1188.321 2.4333 100 10 95.1334 5.1334 2.4333 1188.321 2.4334 100 10 95.1332 5.1332 2.4334 1188.321 2.4335 100 10 95.133 5.133 2.4335 1188.321 2.4336 100 10 95.1328 5.1328 2.4336 1188.321 2.4337 100 10 95.1326 5.1326 2.4337 1188.321 2.4338 100 10 95.1324 5.1324 2.4338 1188.321 2.4339 100 10 95.1322 5.1322 2.4339 1188.321 2.434 100 10 95.132 5.132 2.434 1188.321 2.4341 100 10 95.1318 5.1318 2.4341 1188.321 2.4342 100 10 95.1316 5.1316 2.4342 1188.321 2.4343 100 10 95.1314 5.1314 2.4343 1188.321 2.4344 100 10 95.1312 5.1312 2.4344 1188.321 2.4345 100 10 95.131 5.131 2.4345 1188.321 2.4346 100 10 95.1308 5.1308 2.4346 1188.321 2.4347 100 10 95.1306 5.1306 2.4347 1188.321 2.4348 100 10 95.1304 5.1304 2.4348 1188.321 2.4349 100 10 95.1302 5.1302 2.4349 1188.321 2.435 100 10 95.13 5.13 2.435 1188.321 2.4351 100 10 95.1298 5.1298 2.4351 1188.321 2.4352 100 10 95.1296 5.1296 2.4352 1188.321 2.4353 100 10 95.1294 5.1294 2.4353 1188.321 2.4354 100 10 95.1292 5.1292 2.4354 1188.321 2.4355 100 10 95.129 5.129 2.4355 1188.321 2.4356 100 10 95.1288 5.1288 2.4356 1188.321 2.4357 100 10 95.1286 5.1286 2.4357 1188.321 2.4358 100 10 95.1284 5.1284 2.4358 1188.321 2.4359 100 10 95.1282 5.1282 2.4359 1188.321 2.436 100 10 95.128 5.128 2.436 1188.321

Now we can tell that the cut size makes no difference at all to the volume, therefore I shall not bother to do any more.

2.435/10 = 0.2435, which again makes a proportion of nearly a ¼.

Now I can give a rule to show that to make the maximum volume of a rectangle you must divide the width by 4. This will give the cut size needed to make the maximum volume.

If I understood calculus it could be used to prove my rule.

This student written piece of work is one of many that can be found in our GCSE Comparing length of words in newspapers section.

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