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  • Level: GCSE
  • Subject: Maths
  • Word count: 5614

Open Box Problem

Extracts from this document...

Introduction

Introduction

In this project, I am aiming to:

  1. Determine the size of square cut from any given square sheet of card which makes the volume of an open top box as large as possible.
  2. Determine the size of square cut from any given rectangular sheet of card which makes the volume of the resulting open top box as large as possible

4 squares were cut from the paper (1 from each corner). It was then folded along the lines (see diagram), to make an open top cuboid. Different size squares being cut from the paper each time resulted in a different volume. I spent time ring to calculate the size of the square using trial and improvement.

Firstly, I examined the size of cut that gave the largest volume of open box by using squared paper to test out some different sizes of squares and rectangles. I then used Microsoft Excel spreadsheets to calculate the lengths, depths and widths to give me the volume of the open box. I calculated the size of cut that would give me the greatest volume to 3 decimal places.image00.png

image03.pngimage02.pngimage01.pngimage01.png

To create the box, the equal size squares are cut from the four corners of the card, and it is then folded along the dotted lines.image02.png

I then put the resultant data into tables to try and calculate relationships between things such as length and square cut.

...read more.

Middle

8.37

33.26

33.26

8.37

9259.125012

8.38

33.24

33.24

8.38

9259.041888

8.39

33.22

33.22

8.39

9258.938876

8.4

33.2

33.2

8.4

9258.816

8.5

33

33

8.5

9256.5

8.9

32.2

32.2

8.9

9227.876

9

32

32

9

9216

10

30

30

10

9000

11

28

28

11

8624

12

26

26

12

8112

13

24

24

13

7488

14

22

22

14

6776

15

20

20

15

6000

16

18

18

16

5184

17

16

16

17

4352

18

14

14

18

3528

19

12

12

19

2736

20

10

10

20

2000

21

8

8

21

1344

22

6

6

22

792

23

4

4

23

368

24

2

2

24

96

25

0

0

25

0

image15.png

Volume in cm3, square cut in cm

Square Formula Conclusions

When X is the size of square cut from each corner, and A is the original length/width of square, then the formula to give the maximum volume is:

X=A/6

So, if you want to find the largest possible volume for an open top square, you must first divide the width/length by 6 to give you the optimum cut, and then carry out the method of formula described in the intoduction – a-(x*2), and then length*width*depth to give you the volume with the optimum cut.

size of square (cm2) (a)

size of cut (cm2) (x)

volume (cm3)

5

0.8333

9.259259248

8

1.333

37.92592415

10

1.6667

74.07407405

50

8.333

9259.259248

By using X=A/6, I should be able to work out the optimum cut and largest volume for any square. So, if I had a square dimensions 20cm x 20cm, I can calculate the optimum cut and give the largest possible volume:

        X=A/6

        X=20/6

X=3.33333cm

This makes it easier for me to calculate the optimum cut, without going to the trouble of finding it on Exel Spreadsheets, which can be quite time consuming.

This means that the largest possible volume of a 20cm x 20cm square is:

        V= X (A-2X) (A-2X)

           = 3.333 (20-(2 x 3.333)) (20-(2 x 3.333))

= 592.5925926 cm3

Investigation 2 – Rectangle

Introduction

Unlike finding the optimum cut out from squares, rectangles will be a lot harder as the optimum cut will be with two variables; length and width, rather than just one. One approach to this problem is looking at the optimum cuts of rectangles whose sides follow a general rule, for example the length to width ratio could be 1: 2 or 1:3. I will then compare and see in which direction the relationships progress in relation to the results for squares.

Ratio 1:4

Rectangular Card Dimensions 3cm x 12cm

3 x 12

X cm

Width cm

Length cm

X cm

Volume cm3

0

3

12

0

0

0.5

2

11

0.5

11

0.55

1.9

10.9

0.55

11.3905

0.6

1.8

10.8

0.6

11.664

0.65

1.7

10.7

0.65

11.8235

0.66

1.68

10.68

0.66

11.84198

0.67

1.66

10.66

0.67

11.85605

0.68

1.64

10.64

0.68

11.86573

0.69

1.62

10.62

0.69

11.87104

0.691

1.618

10.618

0.691

11.87133

0.692

1.616

10.616

0.692

11.87158

0.693

1.614

10.614

0.693

11.87178

0.694

1.612

10.612

0.694

11.87194

0.695

1.61

10.61

0.695

11.87206

0.696

1.608

10.608

0.696

11.87213

0.697

1.606

10.606

0.697

11.87217

0.698

1.604

10.604

0.698

11.87215

0.699

1.602

10.602

0.699

11.8721

0.7

1.6

10.6

0.7

11.872

0.75

1.5

10.5

0.75

11.8125

0.8

1.4

10.4

0.8

11.648

0.85

1.3

10.3

0.85

11.3815

0.9

1.2

10.2

0.9

11.016

0.95

1.1

10.1

0.95

10.5545

1

1

10

1

10

1.5

0

9

1.5

0

image16.png

Square cut in cm, volume in cm3

Rectangular Card Dimensions 5cm x 20cm

5 x 20

X cm

Length cm

Width cm

X cm

Volume cm3

0

5

20

0

0

0.5

4

19

0.5

38

1

3

18

1

54

1.1

2.8

17.8

1.1

54.824

1.11

2.78

17.78

1.11

54.86552

1.12

2.76

17.76

1.12

54.89971

1.13

2.74

17.74

1.13

54.92659

1.14

2.72

17.72

1.14

54.94618

1.15

2.7

17.7

1.15

54.9585

1.16

2.68

17.68

1.16

54.96358

1.161

2.678

17.678

1.161

54.9637

1.162

2.676

17.676

1.162

54.96373

1.163

2.674

17.674

1.163

54.9637

1.164

2.672

17.672

1.164

54.9636

1.165

2.67

17.67

1.165

54.96342

1.166

2.668

17.668

1.166

54.96317

1.167

2.666

17.666

1.167

54.96285

1.168

2.664

17.664

1.168

54.96245

1.169

2.662

17.662

1.169

54.96199

1.17

2.66

17.66

1.17

54.96145

1.18

2.64

17.64

1.18

54.95213

1.19

2.62

17.62

1.19

54.93564

1.2

2.6

17.6

1.2

54.912

1.3

2.4

17.4

1.3

54.288

1.4

2.2

17.2

1.4

52.976

1.5

2

17

1.5

51

2

1

16

2

32

2.5

0

15

2.5

0

image17.png

Square cut in cm, volume in cm3

Rectangular Card Dimensions 10cm x 40cm

10 x 40

X cm

Length cm

Width cm

X cm

Volume cm

0

10

40

0

0

0.5

9

39

0.5

175.5

1

8

38

1

304

1.5

7

37

1.5

388.5

2

6

36

2

432

2.1

5.8

35.8

2.1

436.044

2.2

5.6

35.6

2.2

438.592

2.3

5.4

35.4

2.3

439.668

2.31

5.38

35.38

2.31

439.6956

2.32

5.36

35.36

2.32

439.7087

2.321

5.358

35.358

2.321

439.7092

2.322

5.356

35.356

2.322

439.7096

2.323

5.354

35.354

2.323

439.7098

2.324

5.352

35.352

2.324

439.7099

2.325

5.35

35.35

2.325

439.7098

2.33

5.34

35.34

2.33

439.7073

2.34

5.32

35.32

2.34

439.6916

2.35

5.3

35.3

2.35

439.6615

2.4

5.2

35.2

2.4

439.296

2.5

5

35

2.5

437.5

3

4

34

3

408

3.5

3

33

3.5

346.5

4

2

32

4

256

4.5

1

31

4.5

139.5

5

0

30

5

0

image18.png

Square cut in cm, volume in cm3

x

a

b

 0.697

3

12

1.162

5

20

2.324

10

40

3/0.697

4.304160689

5/1.162

4.30292599

10/2.324

4.30292599

 When the ratio is 1:4, the formula for the optimum cut is always A divided by 4.3

X=A/4.30

Ratio 1:3

Rectangular Card Dimensions 3cm x 9cm

3 x 9

X cm

Width cm

Length cm

X cm

Volume cm3

0

3

9

0

0

0.5

2

8

0.5

8

0.6

1.8

7.8

0.6

8.424

0.61

1.78

7.78

0.61

8.447524

0.62

1.76

7.76

0.62

8.467712

0.63

1.74

7.74

0.63

8.484588

0.64

1.72

7.72

0.64

8.498176

0.65

1.7

7.7

0.65

8.5085

0.66

1.68

7.68

0.66

8.515584

0.67

1.66

7.66

0.67

8.519452

0.671

1.658

7.658

0.671

8.519663

0.672

1.656

7.656

0.672

8.519842

0.673

1.654

7.654

0.673

8.519989

0.674

1.652

7.652

0.674

8.520104

0.675

1.65

7.65

0.675

8.520188

0.676

1.648

7.648

0.676

8.520239

0.677

1.646

7.646

0.677

8.520259

0.678

1.644

7.644

0.678

8.520247

0.679

1.642

7.642

0.679

8.520203

0.68

1.64

7.64

0.68

8.520128

0.69

1.62

7.62

0.69

8.517636

0.7

...read more.

Conclusion

Conclusion

Underneath I have produced a table showing the squares and rectangles together.

Ratio

Optimum Cut Formula

01:01

Length(cm)/6

01:02

Length(cm)/4.73

01:03

Length(cm)/4.43

01:04

Length(cm)/4.3

For a piece of card that is square in shape I came up with the formula:

x = original length (cm)/6      

After that I investigated a rectangular piece of card which had a length that was twice the size of the width. For this I came up with the formula:

X = original length (cm)/4.73

I then investigated a rectangular piece of card which had a length 3 times the size of the width. For this piece of card I came up with the formula:

 X = original length (cm)/4.43            

After that I investigated a rectangular piece of card which had a length 4 times bigger than the width. For this particular piece of card I came up with the formula:

X= original length (cm)/4.3

From this I conclude that as the ratio of length to width gets smaller (1:3 becomes 1:2), the rectangle becomes closer and closer in its dimensions to a square (1:1). Therefore, the denominator (imagine x=original length (cm)), will become closer and closer to that of the square(6).                                            4.3image04.png

image12.png

Numberimage05.png

needed

image06.png

ratio

As you can see by the last dozen pages or so, we can see that there is no formula to give you the optimum cut for any size rectangle, there is only a different formula for every ratio. Also, if you look at the line graph above, you can see that my theory that the number needed to divide the original length by to give you optimum X gradually gets closer to that of the ratio for a square (1:1), which is 6.

...read more.

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