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• Level: GCSE
• Subject: Maths
• Word count: 2162

# The Open Box Problem

Extracts from this document...

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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