# 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 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 |

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

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

Conclusion

x = width of rectangular shaped piece of card

x/4.7 = side of cut-out corner

Volume = Length x Width x Height

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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

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

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

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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

This student written piece of work is one of many that can be found in our GCSE Open Box Problem section.

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