# Maths Coursework - The Open Box Problem

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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 |

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 | 590.364 |

3.2 | 13.6 | 13.6 | 591.872 |

3.3 | 13.4 | 13.4 | 592.548 |

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 |

Cut Out x | Width 20-2x | Length 20-2x | Volume |

3.31 | 13.38 | 13.38 | 592.571 |

3.32 | 13.36 | 13.36 | 592.585 |

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 | 592.593 |

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 |

Middle

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 | Volume |

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 | Volume |

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 | Volume |

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.

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 is, 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 | 14812 |

8 | 44 | 44 | 15488 |

9 | 42 | 42 | 15876 |

10 | 40 | 40 | 16000 |

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:

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.

This is the graph for the 150:30 rectangle.

This is the graph for the 200:40 rectangle.

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:

This student written piece of work is one of many that can be found in our GCSE Gradient Function section.

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