# Investigate the size of the cut out square, from any square sheet of card, which makes an open box of the largest volume.

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Introduction

Maths Coursework- The Open Box Problem Mary-Louise Duffy 11R Part 1 I am going to investigate the size of the cut out square, from any square sheet of card, which makes an open box of the largest volume. Firstly, I am going to do two investigations, using exact numbers, of this box so I can establish a connection between the size of the sheet and the size of the cut-out square. Then if I found out a similarity between the two, then I will use another example to prove my theory. This is a diagram of the of the box: C= Cut-out square size X= original length of card This is a formula to work out any volume, and I'm going to change it, so it's shows how to work out the volume of the card above: Volume = length x width x height V= (X-2C) x (X-2C) x C V= (X - 2C) Squared x C So the final formula to work out the volume of this box is: V= C(X-2C) squared This is my first investigation: 30cm by 30cm piece of card. ...read more.

Middle

divided by 3cm = 6 - That's same number I got in the last investigation, when I divided the length of the piece of card by the size of the square cut-out. My calculations tell me that the cut-out could be a sixth of the length of the card. (X) So my formula for working out the cut-out is Now, I'm going to prove my theory by doing a calculation. 30cm divided by 6 = C 5 = C My theory is correct, because in my first investigation, the 30cm by 30cm piece of card, the highest volume was produced by 5cm x 5cm square cut-out. Task 2 I'm going to investigate the size of the cut-out square, from any sized rectangular piece of card that makes an open box of the largest volume. As I did in the previous task, I will do two investigations and if I come up with a theory, I will prove it by doing another set of calculations. As I also did in the previous task, I will try to find a connection between the size of the square cut-out and the size of the rectangular piece of card it is being cut from. ...read more.

Conclusion

This is my third investigation: 40cm by 20cm piece of card In the table above, you can see the largest volume produced (1539.6cm cubed 2.d.p.) was by the 4.2222c, (recurring) sized square cut-out. As you may see in the table above, I checked around what I thought was the largest volume, and found out it wasn't the highest. So I kept on checking the decimals, until I realised that the volume would keep on continuously increasing as more 2's were added on the end of 4.222. Now I'm going to do my last set of calculations. 40cm divided by 4.22 (recurring)= 16.140351 20cm divided by 4.22 (recurring)= 4.7368421 4.22 (recurring) divided by 40cm= 0.1055555 4.22 (recurring) divided by 20cm= 0.2111111 When I first started this task, I thought there must be some kind of connection or relationship between the maximum volume square cut-out and the size of the piece of the card, as on the rectangle was only double the square. But I can't find it. But what I did notice was that on investigations 2 + 3, when you divided the size of the square cut-out by the width of the piece of card and you got the same number which was 0.1055555. So there must be some kind of connection but I am unable to find. ?? ?? ?? ?? ...read more.

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