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  • Level: GCSE
  • Subject: Maths
  • Word count: 2162

The Open Box Problem

Extracts from this document...

Introduction

The Open Box Problem 

An open box is to be made from a sheet of card as shown below. The corner squares are to be cut-off.

The card is then folded along the dotted lines to make the box.

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Investigation 1 – Square shaped pieces of card

Aim – To find the length of the cut-out corners that gives the maximum volume for the open box formed for any sized piece of square card. The length of the square cut will be to 3 significant figures of accuracy.

Method – I will investigate what length of cut-out corners will give the largest volume ofr square pieces of card with dimensions 12 x 12, 18 x 18, 24 x 24 and 30 x 30.

NOTE – when ‘small side’ is mentioned, it refers to the size of the cut-out corners.

When ‘Length’, ‘Width’ and ‘Height’ are mentioned, they refer to the dimensions of the open box.

When ‘Volume’ is mentioned, it refers to the volume of the open box. 

Rows in Italics are those which contain the correct cut-out corner size for the maximum volume of the open box.

Square piece of card with dimensions 12 x 12

Small Side

Volume

Length

Width

Height

1

100

10

10

1

2

128

8

8

2

3

108

6

6

3

2.1

127.764

7.8

7.8

2.1

2.2

127.072

7.6

7.6

2.2

2.3

125.948

7.4

7.4

2.3

2.4

124.416

7.2

7.2

2.4

2.5

122.5

7

7

2.5

1.9

127.756

8.2

8.2

1.9

1.8

127.008

8.4

8.4

1.8

1.95

127.9395

8.1

8.1

1.95

1.99

127.9976

8.02

8.02

1.99

...read more.

Middle

Square piece of card with  dimensions  of 30 x 30

Small Side

Volume

Length

Width

Height

3

1728

24

24

3

4

1936

22

22

4

5

2000

20

20

5

4.5

1984.5

21

21

4.5

4.6

1990.144

20.8

20.8

4.6

4.7

1994.492

20.6

20.6

4.7

4.8

1997.568

20.4

20.4

4.8

4.9

1999.396

20.2

20.2

4.9

4.91

1999.5111

20.18

20.2

4.91

4.95

1999.8495

20.1

20.1

4.95

4.99

1999.994

20.02

20

4.99

Graph comparing the length of Small Side to the Volume for a square shaped piece of card with dimensions 30 x 30

image03.png
 


Formula to give the maximum open box  volume for square shaped pieces of card

x = Side of square shaped piece of card

x/6 = side of cut-out corner

Volume = Length x Width x Height

_______________________________________________________

_______________________________________________________

_______________________________________________________

_______________________________________________________

_______________________________________________________

_______________________________________________________

________________________________________________________

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Investigation 2 –  Rectangular shaped pieces of card

Aim- To find the length of the cut-out corner squares that give the maximum open box volume for rectangular pieces of card of different sizes. The length of the cut-out corner squares will be to 3 significant figures of accuracy.

Method- I will investigate the size of the corner cut-out squares that give the largest open box volume for rectangular shaped pieces of card  that have the width to length ratio of 1:2, those being 12 x 24, 24 x 48 and 48 x 96. I will then produce a formula for the maximum open box volume, for all rectangles that have width to length ratio’s of 1:2. I will also investigate the size of

...read more.

Conclusion

x = width  of rectangular shaped piece of card

x/4.7 = side of cut-out corner

Volume = Length x Width x Height

_______________________________________________________

_______________________________________________________

_______________________________________________________

_______________________________________________________

_______________________________________________________

_______________________________________________________

________________________________________________________

Rectangular piece of card with dimensions 5 x 15

Small Side

Volume

Length

Width

Height

1

39

13

3

1

2

22

11

1

2

1.5

36

12

2

1.5

1.4

37.576

12.2

2.2

1.4

1.3

38.688

12.4

2.4

1.3

1.2

39.312

12.6

2.6

1.2

1.1

39.424

12.8

2.8

1.1

1.15

39.4335

12.7

2.7

1.15

1.14

39.442176

12.72

2.72

1.14

1.13

39.445588

12.74

2.74

1.13

Graph comparing the length of Small Side to the Volume for a rectangular  shaped piece of card with dimensions   5 x 15

image07.png
 


Rectangular piece of card with dimensions 15 x 45

Small Side

Volume

Length

Width

Height

2

902

41

11

2

3

1053

39

9

3

4

1036

37

7

4

5

875

35

5

5

3.5

1064

38

8

3.5

3.4

1065.016

38.2

8.2

3.4

3.3

1064.448

38.4

8.4

3.3

3.45

1064.7045

38.1

8.1

3.45

3.44

1064.7983

38.12

8.12

3.44

3.43

1064.8764

38.14

8.14

3.43

3.42

1064.9388

38.16

8.16

3.42

3.41

1064.9853

38.18

8.18

3.41

Graph comparing the length of Small Side to the Volume for a rectangular  shaped piece of card with dimensions   15 x 45

image08.png
 


Rectangular piece of card with dimensions 45 x 135

Small Side

Volume

Length

Width

Height

7

26257

121

31

7

8

27608

119

29

8

9

28431

117

27

9

10

28750

115

25

10

10.5

28728

114

24

10.5

10.4

28741.856

114.2

24.2

10.4

10.3

28751.008

114.4

24.4

10.3

10.2

28755.432

114.6

24.6

10.2

10.1

28755.104

114.8

24.8

10.1

Graph comparing the length of Small Side to the Volume for a rectangular  shaped piece of card with dimensions   15 x 45


 


In each of the above cases the length of the small side that gives the maximum open box volume has been width/4.4

Formula to give the maximum open box  volume for rectangular shaped pieces of card with the ratio of 1:3

x = width  of rectangular shaped piece of card

x/4.4 = side of cut-out corner

Volume = Length x Width x Height

_______________________________________________________

_______________________________________________________

_______________________________________________________

_______________________________________________________

_______________________________________________________

_______________________________________________________

________________________________________________________

----------------------------------------------------------------------------------------------------------------------------

Formula to give the maximum open box  volume for rectangular shaped pieces of card with the any ratio

x = width  of rectangular shaped piece of card

x/                     = side of cut-out corner

Volume = Length x Width x Height

...read more.

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