# Open Box Problem

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Introduction

Introduction

In this project, I am aiming to:

- 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.
- 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.

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.

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

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

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 |

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 |

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 |

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.
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). 4.3 Number needed 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. This student written piece of work is one of many that can be found in our GCSE Open Box Problem section. ## Found what you're looking for?- Start learning 29% faster today
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