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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

...read more.

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

image02.png

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

image03.png

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

image04.png

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

...read more.

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-

image07.png

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.

...read more.

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