# The open box problem

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Introduction

The open box problem

An open box is to be made from a sheet of card.

Identical squares are cut off the four corners of the card as shown below.

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

The aim of this project is to determine the size of the square cut out in any given size rectangle sheet of card with the largest volume.

Task 1:

For any sized square sheet of card, investigate the size of the cut out square which makes an open box of the largest volume.

For this task I will use 20cm*20cm square sheet of card. The formula used to get a volume is V=length x Width x height

Middle

1.9

498.636

20

20

2

512

20

20

2.1

524.244

20

20

2.2

535.392

20

20

2.3

545.468

20

20

2.4

554.496

20

20

2.5

562.5

20

20

2.6

569.504

20

20

2.7

575.532

20

20

2.8

580.608

20

20

2.9

584.756

20

20

3

588

20

20

3.1

590.364

20

20

3.2

591.872

20

20

3.3

592.548

20

20

3.4

592.416

20

20

3.5

591.5

now for more accuracy I will use 3 decimal place and I will concentrating in between 3.332 and 3.334 . this time the results should be more accurate then any other charts.

length | width | height | volume |

20 | 20 | 3.332 | 592.5925 |

20 | 20 | 3.333 | 592.5926 |

20 | 20 | 3.334 | 592.5926 |

20 | 20 | 3.335 | 592.5925 |

20 | 20 | 3.336 | 592.5923 |

20 | 20 | 3.337 | 592.5921 |

20 | 20 | 3.338 | 592.5917 |

20 | 20 | 3.339 | 592.5913 |

20 | 20 | 3.340 | 592.5908 |

20 | 20 | 3.341 | 592.5902 |

from the graph the size 3.333 appears to be the highest which means it is the one size that gives out the largest volume.

To find other size to be cut out on other size of paper, I divide 3.333 by 20 making it 0.16665 and in fraction 1/6.

And that answer helped me to make a formula as 1/6 of the 20 cm makes the highest volume so I made this formula:

Volume= V

Length= L

Width= W

Height= H

As length and width are the same in a square we are only going to use length

- V= (L-2L/6)(L-2L/6) x L/6

Simplified:

- V= (L2 - 2L2/6 - 2L2/6 + 4L/36) x L/6

- V= L3/6 -2L3/36 – 2L3 /36 + 4L2/216

Now I have put the denominator as 216

- V= (4L3 + 36L3 - 12L3 - 12L3 )/216
- V= 40L3 – 24L3 /216
- V= 16L3 /216

Simplified:

- V= 2L3/27

Conclusion

=9259.259cm3

Task2

For any sized rectangular sheet of card, investigate the size of the cut out square which makes an open box of the largest volume.

For this task I will be investigating in rectangular sheet of paper. I will be using 20cm of width and 40cm of length in the ratio of 1:2. as from the square investigation I will be cutting out sizes from 1-10cm on each side.

Length | Width | Height | Volume |

40 | 20 | 1 | 684 |

40 | 20 | 2 | 1152 |

40 | 20 | 3 | 1428 |

40 | 20 | 4 | 1536 |

40 | 20 | 5 | 1500 |

40 | 20 | 6 | 1344 |

40 | 20 | 7 | 1092 |

40 | 20 | 8 | 768 |

40 | 20 | 9 | 396 |

40 | 20 | 10 | 0 |

In this chart the highest volume is 1536cm3 by cutting out 4 cm on each side. It is possible to get more accurate results it will be calculated to 1 decimal place. I will be concentrating between 3 to 5for best results.

Length | Width | Height | Volume |

40 | 20 | 3 | 1428 |

40 | 20 | 3.1 | 1445.964 |

40 | 20 | 3.2 | 1462.272 |

40 | 20 | 3.3 | 1476.948 |

40 | 20 | 3.4 | 1490.016 |

40 | 20 | 3.5 | 1501.5 |

40 | 20 | 3.6 | 1511.424 |

40 | 20 | 3.7 | 1519.812 |

40 | 20 | 3.8 | 1526.688 |

40 | 20 | 3.9 | 1532.076 |

40 | 20 | 4 | 1536 |

40 | 20 | 4.1 | 1538.484 |

40 | 20 | 4.2 | 1539.552 |

40 | 20 | 4.3 | 1539.228 |

40 | 20 | 4.4 | 1537.536 |

40 | 20 | 4.5 | 1534.5 |

40 | 20 | 4.6 | 1530.144 |

40 | 20 | 4.7 | 1524.492 |

40 | 20 | 4.8 | 1517.568 |

40 | 20 | 4.9 | 1509.396 |

40 | 20 | 5 | 1500 |

in this graph and chart data, the highest point is 1539.552cm3 from 4.2 cm cut out. Now I will be calculating the results in three decimal places

This student written piece of work is one of many that can be found in our GCSE Comparing length of words in newspapers section.

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