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  • Level: GCSE
  • Subject: Maths
  • Word count: 3299

The Open Box Problem

Extracts from this document...

Introduction

Math Coursework

The Open Box Problem

An open box is to be made from a sheet of card

Identical squares are to be cut off the four corners of the card as shown in the following diagram

image00.png

Task:

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

In this section of this investigation I am going to investigate using both trial and improvement and spreadsheets which size cutout will give the largest volume of the box.

...read more.

Middle

27.6

4722.912

40

6.3

27.4

4729.788

40

6.4

27.2

4734.976

40

6.5

27

4738.5

40

6.6

26.8

4740.384

40

6.7

26.6

4740.652

40

6.8

26.4

4739.328

40

6.9

26.2

4736.436

40

7

26

4732

S

X

B

V

40

6.6

26.8

4740.38400

40

6.61

26.78

4740.48312

40

6.62

26.76

4740.56611

40

6.63

26.74

4740.63299

40

6.64

26.72

4740.68378

40

6.65

26.7

4740.71850

40

6.66

26.68

4740.73718

40

6.67

26.66

4740.73985

40

6.68

26.64

4740.72653

40

6.69

26.62

4740.69724

40

6.7

26.6

4740.65200

S

X

B

V

40

6.66

26.68

4740.73718

40

6.661

26.678

4740.73817

40

6.662

26.676

4740.73900

40

6.663

26.674

4740.73966

40

6.664

26.672

4740.74017

40

6.665

26.67

4740.74052

40

6.666

26.668

4740.74071

40

6.667

26.666

4740.74073

40

6.668

26.664

4740.74060

40

6.669

26.662

4740.74031

40

6.67

26.66

4740.73985

Percentage Form Of X

S

X

S2

X2

%age X

10

1.666

100

2.775556

2.775556

20

3.333

400

11.10889

2.777222

30

5

900

25

2.777778

40

6.666

1600

44.43556

2.777222

Conclusion

I have found out using my tables that 2.77% of the total area is the best value for X2.

X2=2.77% of S2

To find the percentage value of all of the Xs you need to multiply 2.77 by 4. The 4 is all 4 corners that need to be cut out.

4X2 = (4 x 2.77)% of S2

4X2 = 11.08 % of S2

The Rectangle

image05.png

I will start with a 10 x 5 rectangle

S1

S2

X

B1

B2

V

10

5

1

8

3

24

10

5

2

6

1

12

10

5

3

4

-1

-12

10

5

4

2

-3

-24

10

5

5

0

-5

0

10

5

6

-2

-7

84

10

5

7

-4

-9

252

10

5

8

-6

-11

528

10

5

9

-8

-13

936

10

5

10

-10

-15

1500

S1

S2

X

B1

B2

V

10

5

0

10

5

0

10

5

0.5

9

4

18

10

5

1

8

3

24

10

5

1.5

7

2

21

10

5

2

6

1

12

10

5

2.5

5

0

0

10

5

3

4

-1

-12

10

5

3.5

3

-2

-21

10

5

4

2

-3

-24

10

5

4.5

1

-4

-18

S1

S2

X

B1

B2

V

10

5

1

8

3

24

10

5

1.1

7.8

2.8

24.024

10

5

1.2

7.6

2.6

23.712

10

5

1.3

7.4

2.4

23.088

10

5

1.4

7.2

2.2

22.176

10

5

1.5

7

2

21

10

5

1.6

6.8

1.8

19.584

10

5

1.7

6.6

1.6

17.952

10

5

1.8

6.4

1.4

16.128

10

5

1.9

6.2

1.2

14.136

10

5

2

6

1

12

S1

S2

X

B1

B2

V

10

5

1

8

3

24

10

5

1.01

7.98

2.98

24.0182

10

5

1.02

7.96

2.96

24.03283

10

5

1.03

7.94

2.94

24.04391

10

5

1.04

7.92

2.92

24.05146

10

5

1.05

7.9

2.9

24.0555

10

5

1.06

7.88

2.88

24.05606

10

5

1.07

7.86

2.86

24.05317

10

5

1.08

7.84

2.84

24.04685

10

5

1.09

7.82

2.82

24.03712

10

5

1.1

7.8

2.8

24.024

S1

S2

X

B1

B2

V

10

5

1.05

7.9

2.9

24.0555

10

5

1.051

7.898

2.898

24.05571

10

5

1.052

7.896

2.896

24.05589

10

5

1.053

7.894

2.894

24.05603

10

5

1.054

7.892

2.892

24.05614

10

5

1.055

7.89

2.89

24.05622

10

5

1.056

7.888

2.888

24.05625

10

5

1.057

7.886

2.886

24.05626

10

5

1.058

7.884

2.884

24.05623

10

5

1.059

7.882

2.882

24.05616

10

5

1.06

7.88

2.88

24.05606

20 x 10

S1

S2

X

B1

B2

V

20

10

1

18

8

144

20

10

2

16

6

192

20

10

3

14

4

168

20

10

4

12

2

96

20

10

5

10

0

0

20

10

6

8

-2

-96

20

10

7

6

-4

-168

20

10

8

4

-6

-192

20

10

9

2

-8

-144

20

10

10

0

-10

0

20

10

11

-2

-12

264

S1

S2

X

B1

B2

V

20

10

2

16

6

192

20

10

2.1

15.8

5.8

192.444

20

10

2.2

15.6

5.6

192.192

20

10

2.3

15.4

5.4

191.268

20

10

2.4

15.2

5.2

189.696

20

10

2.5

15

5

187.5

20

10

2.6

14.8

4.8

184.704

20

10

2.7

14.6

4.6

181.332

20

10

2.8

14.4

4.4

177.408

20

10

2.9

14.2

4.2

172.956

20

10

3

14

4

168

...read more.

Conclusion

n="1" rowspan="1">

5.66

192.3393

20

10

2.18

15.64

5.64

192.2969

20

10

2.19

15.62

5.62

192.2478

20

10

2.2

15.6

5.6

192.192

S1

S2

X

B1

B2

V

20

10

2.11

15.78

5.78

192.4497

20

10

2.111

15.778

5.778

192.4499

20

10

2.112

15.776

5.776

192.45

20

10

2.113

15.774

5.774

192.4501

20

10

2.114

15.772

5.772

192.4501

20

10

2.115

15.77

5.77

192.45

20

10

2.116

15.768

5.768

192.4498

20

10

2.117

15.766

5.766

192.4496

20

10

2.118

15.764

5.764

192.4493

20

10

2.119

15.762

5.762

192.4489

20

10

2.12

15.76

5.76

192.4485

30 x 15

S1

S2

X

B1

B2

V

30

15

1

28

13

364

30

15

2

26

11

572

30

15

3

24

9

648

30

15

4

22

7

616

30

15

5

20

5

500

30

15

6

18

3

324

30

15

7

16

1

112

30

15

8

14

-1

-112

30

15

9

12

-3

-324

30

15

10

10

-5

-500

30

15

11

8

-7

-616

S1

S2

X

B1

B2

V

30

15

3

24

9

648

30

15

3.1

23.8

8.8

649.264

30

15

3.2

23.6

8.6

649.472

30

15

3.3

23.4

8.4

648.648

30

15

3.4

23.2

8.2

646.816

30

15

3.5

23

8

644

30

15

3.6

22.8

7.8

640.224

30

15

3.7

22.6

7.6

635.512

30

15

3.8

22.4

7.4

629.888

30

15

3.9

22.2

7.2

623.376

30

15

4

22

7

616

S1

S2

X

B1

B2

V

30

15

3.1

23.8

8.8

649.264

30

15

3.11

23.78

8.78

649.3319

30

15

3.12

23.76

8.76

649.3893

30

15

3.13

23.74

8.74

649.4362

30

15

3.14

23.72

8.72

649.4726

30

15

3.15

23.7

8.7

649.4985

30

15

3.16

23.68

8.68

649.514

30

15

3.17

23.66

8.66

649.5191

30

15

3.18

23.64

8.64

649.5137

30

15

3.19

23.62

8.62

649.498

30

15

3.2

23.6

8.6

649.472

S1

S2

X

B1

B2

V

30

15

3.161

23.678

8.678

649.51496

30

15

3.162

23.676

8.676

649.51583

30

15

3.163

23.674

8.674

649.5166

30

15

3.164

23.672

8.672

649.51726

30

15

3.165

23.67

8.67

649.51782

30

15

3.166

23.668

8.668

649.51827

30

15

3.167

23.666

8.666

649.51862

30

15

3.168

23.664

8.664

649.51887

30

15

3.169

23.662

8.662

649.51901

30

15

3.17

23.66

8.66

649.51905

30

15

3.171

23.658

8.658

649.51899

40 x 20

S1

S2

X

B1

B2

V

40

20

1

38

18

684

40

20

2

36

16

1152

40

20

3

34

14

1428

40

20

4

32

12

1536

40

20

5

30

10

1500

40

20

6

28

8

1344

40

20

7

26

6

1092

40

20

8

24

4

768

40

20

9

22

2

396

40

20

10

20

0

0

40

20

11

18

-2

-396

S1

S2

X

B1

B2

V

40

20

4

32

12

1536

40

20

4.1

31.8

11.8

1538.484

40

20

4.2

31.6

11.6

1539.552

40

20

4.3

31.4

11.4

1539.228

40

20

4.4

31.2

11.2

1537.536

40

20

4.5

31

11

1534.5

40

20

4.6

30.8

10.8

1530.144

40

20

4.7

30.6

10.6

1524.492

40

20

4.8

30.4

10.4

1517.568

40

20

4.9

30.2

10.2

1509.396

40

20

5

30

10

1500

S1

S2

X

B1

B2

V

40

20

4.2

31.6

11.6

1539.552

40

20

4.21

31.58

11.58

1539.5818

40

20

4.22

31.56

11.56

1539.5978

40

20

4.23

31.54

11.54

1539.5999

40

20

4.24

31.52

11.52

1539.5881

40

20

4.25

31.5

11.5

1539.5625

40

20

4.26

31.48

11.48

1539.5231

40

20

4.27

31.46

11.46

1539.4699

40

20

4.28

31.44

11.44

1539.403

40

20

4.29

31.42

11.42

1539.3224

40

20

4.3

31.4

11.4

1539.228

S1

S2

X

B1

B2

V

40

20

4.22

31.56

11.56

1539.5978

40

20

4.221

31.558

11.558

1539.5986

40

20

4.222

31.556

11.556

1539.5993

40

20

4.223

31.554

11.554

1539.5999

40

20

4.224

31.552

11.552

1539.6003

40

20

4.225

31.55

11.55

1539.6006

40

20

4.226

31.548

11.548

1539.6007

40

20

4.227

31.546

11.546

1539.6007

40

20

4.228

31.544

11.544

1539.6006

40

20

4.229

31.542

11.542

1539.6003

40

20

4.23

31.54

11.54

1539.5999

Percentage Form of X

S1

S2

X

S1 x S2

X2

%age of X2

10

5

1.057

50

1.117249

2.114

20

10

2.113

200

4.464769

1.0565

30

15

3.17

450

10.0489

0.7044444

40

20

4.226

800

17.85908

0.52825

Conclusion ...read more.

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