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  • Level: GCSE
  • Subject: Maths
  • Word count: 2528

Maths Coursework - The Open Box Problem

Extracts from this document...

Introduction

Maths Coursework – The Open Box Problem

In this investigation we are asked to determine the size of the square cut out which makes the volume of the box as large as possible for any given rectangular sheet of card. This investigation is dived up into 2 parts; Part 1 and Part 2.

Part 1

For this part we are asked to determine for any sized square sheet of card, investigate the size of the cut out square which makes an open box of the largest volume.

Part 2

For this part we are asked to determine for any sized rectangular sheet of card, investigate the size of the cut out square which makes an open box of the largest volume.

Part 1

I decided to construct a table to investigate what length of cut out would give the largest volume.

This is the table I constructed for a 10 by 10 square using the program Microsoft Excel.

20 by 20 square

Cut Out x

Width 20-2x

Length 20-2x

Volume

1

18

18

324

2

16

16

512

3

14

14

588

4

12

12

576

5

10

10

500

6

8

8

384

7

6

6

252

8

4

4

128

9

2

2

36

10

0

0

0

image00.png

image01.png

I have now found out that in a 10 by 10 square the cut out which gives the largest volume is between 3 and 4 but more towards 3 since 3 gives a larger volume. To obtain a more accurate result I will zoom in the shaded region.

Cut Out x

Width 20-2x

Length 20-2x

Volume

3.1

13.8

13.8

590.364image05.png

3.2

13.6

13.6

591.872

3.3

13.4

13.4

592.548image01.png

3.4

13.2

13.2

592.416

3.5

13

13

591.5

3.6

12.8

12.8

589.824

3.7

12.6

12.6

587.412

3.8

12.4

12.4

584.288

3.9

12.2

12.2

580.476

4

12

12

576

image06.png

Cut Out x

Width 20-2x

Length 20-2x

Volume

3.31

13.38

13.38

592.571

3.32

13.36

13.36

592.585image03.pngimage01.png

3.33

13.34

13.34

592.592

3.34

13.32

13.32

592.591

3.35

13.3

13.3

592.582

3.36

13.28

13.28

592.564

3.37

13.26

13.26

592.539

3.38

13.24

13.24

592.506

3.39

13.22

13.22

592.465

3.4

13.2

13.2

592.416

I will zoom in again to increase the degree of accuracy in finding out what length of cut out gives the largest volume.

Cut Out x

Width 20-2x

Length 20-2x

Volume

3.331

13.338

13.338

592.592

3.332

13.336

13.336

592.593image03.pngimage01.png

3.333

13.334

13.334

592.593

3.334

13.332

13.332

592.593

3.335

13.33

13.33

592.592

3.336

13.328

13.328

592.592

3.337

13.326

13.326

592.592

3.338

13.324

13.324

592.592

3.339

13.322

13.322

592.591

3.34

13.32

13.32

592.591

...read more.

Middle

10

10

10

1000

I have now found out that in a 10 by 10 square the cut out which gives the largest volume is between 5 and 6 but more towards 5 since 5 gives a larger volume. To obtain a more accurate result I will zoom in the shaded region.

Cut Out x

Width 30-2x

Length 10-2x

Volumeimage03.pngimage01.png

5.1

19.8

19.8

1999.404

5.2

19.6

19.6

1997.632

5.3

19.4

19.4

1994.708

5.4

19.2

19.2

1990.656

5.5

19

19

1985.5

5.6

18.8

18.8

1979.264

5.7

18.6

18.6

1971.972

5.8

18.4

18.4

1963.648

5.9

18.2

18.2

1954.316

6

18

18

1944

After zooming in I have found out that a value between 5.1 and 5.2 for the cut out gives the highest volume. I have not found a pattern in the 30 by 30 square cut outs like I did with the 20 by 20 square. I have given the volume to 3 decimal place since it is a sensible degree of accuracy although the cut out is only to 1 decimal place therefore I will zoom in further to obtain a more accurate result.

Cut Out x

Width 30-2x

Length 10-2x

Volumeimage03.pngimage01.png

5.11

19.78

19.78

1999.279

5.12

19.76

19.76

1999.143

5.13

19.74

19.74

1998.995

5.14

19.72

19.72

1998.835

5.15

19.7

19.7

1998.664

5.16

19.68

19.68

1998.48

5.17

19.66

19.66

1998.286

5.18

19.64

19.64

1998.079

5.19

19.62

19.62

1997.861

5.2

19.6

19.6

1997.632

I will zoom in again to increase the degree of accuracy in finding out what length of cut out gives the largest volume.

Cut Out x

Width 30-2x

Length 10-2x

Volumeimage01.pngimage03.png

5.111

19.778

19.778

1999.266

5.112

19.776

19.776

1999.253

5.113

19.774

19.774

1999.24

5.114

19.772

19.772

1999.226

5.115

19.77

19.77

1999.213

5.116

19.768

19.768

1999.199

5.117

19.766

19.766

1999.185

5.118

19.764

19.764

1999.171

5.119

19.762

19.762

1999.157

5.12

19.76

19.76

1999.143

This is a graph I produced of the program Autograph. Like the previous graphs  I have made it bold so that it is easier to interpret and I have also added the maximum value in a pink text box and the equation of the graph in a blue text box.

image08.pngimage04.png

After choosing 3 different cut outs for the square I constructed a general table.

Size of Square

Length of cut out which maximises volume

10 by 20

1.6 recurring

20 by 20

3.3 recurring

30 by 30

5

After analyzing the results I discovered a relationship between the length of the cut out and the size of the square and was able to derive a general expression. For an n by n square the cut out which maximises the volume isimage19.png, where n is the length of the square.

I am now going to investigate if the general formula would work with a square with any size cut out. I am going to pick a random number such as 180 by 180.

60 by 60 square

Cut Out x

Width 60-2x

Length 60-2x

Volume

1

58

58

3364

2

56

56

6272

3

54

54

8748

4

52

52

10816

5

50

50

12500

6

48

48

13824

7

46

46

14812image03.png

8

44

44

15488

9

42

42

15876image01.png

10

40

40

16000

...read more.

Conclusion

Size of Rectangles

Length of cut out which maximises volume

80 by 20

4.648

120 by 30

6.972

160 by 40

9.296

After analysing the table of results I discovered that dividing the length of the cut out by the smallest value of the rectangle equals the same final answer for each size in the 4:1 ratio of rectangles. For example:

  • image20.png
  • image21.png
  • image22.png

Therefore for an n by 4n rectangle the cut out which maximises the volume is 0.2324n.

These are the 3 cut out sizes I used for the 5:1 ratio rectangle.

  • 100:20
  • 150:30
  • 2000:40

This is the graph for the 100:20 rectangle.

image23.png

This is the graph for the 150:30 rectangle.

image24.png

This is the graph for the 200:40 rectangle.

image25.png

I will now try to find out a general rule for the 5:1 ratio rectangles. I have constructed a table for this. To find a general formula that maximises the volume, I have to first find a pattern between the results. The length of cut out which maximises volume is the x value on the maximum point on the graph.

Size of Rectangles

Length of cut out which maximises volume

100 by 20

4.725

150 by 30

7.087

200 by 40

9.449

After analysing the table of results I discovered that dividing the length of the cut out by the smallest value of the rectangle equals the same final answer for each size in the 5:1 ratio of rectangles. For example:

  • image26.png
  • image27.png
  • image28.png

...read more.

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