# First Problem,The Open Box Problem

Extracts from this document...

Introduction

In this investigation, I will be investigating the maximum volume, which can be made from a certain size square piece of card, with different size sections cut from their corners. The types of cubes I will be using are all open topped boxes.

The size sections that I will be cutting from the square piece of card will all be the same size. The section sizes will go up to half of the original size of card. I will only go up to this size, because it is physically impossible to cut square sections, with sides over half the length of the original shape.

During this investigation, I will not account for the ‘tabs’, which would normally be needed to hold the box sides together.

Middle

2.8

2.8

4.704

0.7

0.7

2.6

2.6

4.732

0.8

0.8

2.4

2.4

4.608

0.9

0.9

2.2

2.2

4.356

1.0

1.0

2.0

2.0

4.000

1.1

1.1

1.8

1.8

3.564

1.2

1.2

1.6

1.6

3.072

1.3

1.3

1.4

1.4

2.548

1.4

1.4

1.2

1.2

2.016

1.5

1.5

1.0

1.0

1.500

1.6

1.6

0.8

0.8

1.024

1.7

1.7

0.6

0.6

0.612

1.8

1.8

0.4

0.4

0.288

1.9

1.9

0.2

0.2

0.076

2.0

5cm by 5cm, piece of square card

Length of the section (cm) | Height of the section (cm) | Depth of the section (cm) | Width of the section (cm) | Volume of the cube (cm3) | |

0.1 | 0.1 | 4.8 | 4.8 | 2.304 | |

0.2 | 0.2 | 4.6 | 4.6 | 4.232 | |

0.3 | 0.3 | 4.4 | 4.4 | 5.808 | |

0.4 | 0.4 | 4.2 | 4.2 | 7.056 | |

0.5 | 0.5 | 4.0 | 4.0 | 8.000 | |

0.6 | 0.6 | 3.8 | 3.8 | 8.664 | |

0.7 | 0.7 | 3.6 | 3.6 | 9.072 | |

0.8 | 0.8 | 3.4 | 3.4 | 9.248 | |

0.9 | 0.9 | 3.2 | 3.2 | 9.216 | |

1.0 | 1.0 | 3.0 | 3.0 | 9.000 | |

1.1 | 1.1 | 2.8 | 2.8 | 8.624 | |

1.2 | 1.2 | 2.6 | 2.6 | 8.112 | |

1.3 | 1.3 | 2.4 | 2.4 | 7.488 | |

1.4 | 1.4 | 2.2 | 2.2 | 6.776 | |

1.5 | 1.5 | 2.0 | 2.0 | 6.000 | |

1.6 | 1.6 | 1.8 | 1.8 | 5.184 | |

1.7 | 1.7 | 1.6 | 1.6 | 4.352 | |

1.8 | 1.8 | 1.4 | 1.4 | 3.528 | |

1.9 | 1.9 | 1.2 | 1.2 | 2.736 | |

2.0 | 2.0 | 1.0 | 1.0 | 2.000 | |

2.1 | 2.1 | 0.8 | 0.8 | 1.344 | |

2.2 | 2.2 | 0.6 | 0.6 | 0.792 | |

2.3 | 2.3 | 0.4 | 0.4 | 0.368 | |

2.4 | 2.4 | 0.2 | 0.2 | 0.096 | |

2.5 |

6cm by 6cm, piece of square card

Length of the section (cm) | Height of the section (cm) |

Conclusion

So if A = the size of the side I start with, and X = the size of the cut out, then my results can be described by the following equation:

X = 1/6A

To prove it further, I have decided to test my rule with one larger number, 100cm x 100cm. To test this equation, on the very large square gives the results below:

X = 1/6100

## X = 16.667 cm

VOL. = X ( A - 2X ) ( B - 2X )

= 16.667 ( 100 - 2 x 16.667 ) ( 100 - 2 x 16.667 )

= 16.667 ( 100 - 33.334 ) ( 100 - 33.334 )

= 16.667 x 66.666 x 66.666

= 74074.074 cm3

If I make X slightly bigger or smaller, then the volume should decrease. I will use X = 16.6, and X = 16.75.

Vol. = X ( A - 2X ) ( B - 2X )

= 16.6 ( 100 - 33.2 ) ( 100 - 33.2 )

= 16.6 x 66.8 x 66.8

= 74073.184

Vol. = X ( A - 2X ) ( B - 2X )

= 16.75 ( 100 - 33.5 ) ( 100 -33.5 )

= 16.75 x 66.5 x 66.5

= 74072.687

These are both smaller than my predicted value, and therefore my prediction is correct.

This student written piece of work is one of many that can be found in our GCSE Number Stairs, Grids and Sequences section.

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