# To find out what size squares must be cut out of the corners of square and rectangle pieces of card to give the box it would create its optimum volume.

Extracts from this document...

Introduction

As a class we were given a mathematical problem to solve as our first bit of maths coursework for GCSE, it is called the open box problem. We have been asked to find out what size square we would have to cut out the corner of a piece of card, either square or rectangular, to find the optimum volume of that box. I will start at 1cm³ Square and work up in 1cm³.

I would expect the optimum volume to most of the time be in the decimal places so I will have to look at all the decimal places between the highest volume I have and the highest one on one of its sides. During this experiment I will be looking for relationships between the length of the side and the volume of the box.

## Aim:- To find out what size squares must be cut out of the corners

Middle

2.6=120.224

2.7=117.612

### Optimum Volume=2

#### Mini Summary

I don’t think I have enough results so I’m going to do a 15*15 to increase my results.

15*15 Square

Height (cm²) | 1 | 2 | 3 | 4 | 5 | 6 | 7 |

Volume (cm³) | 169 | 242 | 243 | 196 | 50 | 36 | 7 |

2.1=244.944

2.2=247.192

2.3=248.768

2.4=249.696

2.5=250 Optimum Volume= 2.5 = 250

2.6=249.704

2.7=248.832

2.8=247.408

2.9=245.456

The diagram below will help you to understand my formulas and equations.

10cm (x)

10cm (x)

To find the base area of the square you use the equation (x-2h)(x-2h).

Take h as 1cm

(10-2*1)(10-2*1)=64cm²

To find the size of square to cut out of the corner on a square to find the optimum volume use the formula:- cut-out = Length of side / 6

Conclusion

The one similarity I found between the squares and rectangles comes from looking at the graphs I have from the results. Looking at the graphs I can see that they are similar in shape. They all peak in roughly the same area and then begin to fall quite rapidly, this is what I said was what I thought would happen in my prediction.

With the squares I can see from the ones I have looked at that the squares all peak in the high ones and the low twos. Unlike with the rectangles I found a formula I can use with all squares. It tells me what size I should make the squares on the corners for any size square

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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