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• Level: GCSE
• Subject: Maths
• Word count: 5227

# The Open Box Problem

Extracts from this document...

Introduction

The Open Box Problem

Maths Coursework

Jawhari Thomas 11.5

An open box is a box missing 1 of its six surfaces it can be made using either a squared or rectangular sheet of card. Identical squares are cut from the four corners of the card this creates the height of the box it is then folded as shown below. The card is folded along the dotted lines to form the box.

Stage 1                         Stage 2                Stage 3

The aim of this exercise is to find the formulae that will enable someone determine the size of the squares cut from the corners of the sheet of card to give the greatest the volume of the box.

I am going to begin the investigation using squares, as this will most probably be easiest. I won’t build the boxes I am going to use simple mathematics to work out the volumes.   Firstly I am going to use an example of how I will carry out the experiment using a square 20cm in length. Using this sized length will allow me to only cut off each corner up to 9.9cm as otherwise I will cause me to run out of card. I am going to begin by looking at cutting the squares off as whole numbers. To find the volume of any box we must use the formula:

V = L * W * H

When:

V is Volume

## L is Length

W is Width and

H is Height

Middle

704

30x30

12

24

6

6

432

30x30

13

26

4

4

208

30x30

14

28

2

2

56

30x30

15

30

0

0

0

 30x30 5.1 10.2 19.8 19.8 1999.4 30x30 5.2 10.4 19.6 19.6 1997.63 30x30 5.3 10.6 19.4 19.4 1994.71 30x30 5.4 10.8 19.2 19.2 1990.66 30x30 5.5 11 19 19 1985.5 30x30 5.6 11.2 18.8 18.8 1979.26 30x30 5.7 11.4 18.6 18.6 1971.97 30x30 5.8 11.6 18.4 18.4 1963.65 30x30 5.9 11.8 18.2 18.2 1954.32

8x8

 8x8 1 2 6 6 36 8x8 2 4 4 4 32 Max 8x8 3 6 2 2 12 8x8 4 8 0 0 0
 8x8 2.1 4.2 3.8 3.8 30.324 8x8 2.2 4.4 3.6 3.6 28.512 8x8 2.3 4.6 3.4 3.4 26.588 8x8 2.4 4.8 3.2 3.2 24.576 8x8 2.5 5 3 3 22.5 8x8 2.6 5.2 2.8 2.8 20.384 8x8 2.7 5.4 2.6 2.6 18.252 8x8 2.8 5.6 2.4 2.4 16.128 8x8 2.9 5.8 2.2 2.2 14.036

15x15

 15x15 1 2 13 13 169 15x15 2 4 11 11 242 15x15 3 6 9 9 243 Max 15x15 4 8 7 7 196 15x15 5 10 5 5 125 15x15 6 12 3 3 54 15x15 7 14 1 1 7
 15x15 2.1 4.2 10.8 10.8 244.944 15x15 2.2 4.4 10.6 10.6 247.192 15x15 2.3 4.6 10.4 10.4 248.768 15x15 2.4 4.8 10.2 10.2 249.696 15x15 2.5 5 10 10 250 Max 15x15 2.6 5.2 9.8 9.8 249.704 15x15 2.7 5.4 9.6 9.6 248.832 15x15 2.8 5.6 9.4 9.4 247.408 15x15 2.9 5.8 9.2 9.2 245.456 15x15 3 6 9 9 243 15x15 3.1 6.2 8.8 8.8 240.064 15x15 3.2 6.4 8.6 8.6 236.672 15x15 3.3 6.6 8.4 8.4 232.848 15x15 3.4 6.8 8.2 8.2 228.616 15x15 3.5 7 8 8 224 15x15 3.6 7.2 7.8 7.8 219.024 15x15 3.7 7.4 7.6 7.6 213.712 15x15 3.8 7.6 7.4 7.4 208.088 15x15 3.9 7.8 7.2 7.2 202.176

25x25

 25x25 1 2 23 23 529 25x25 2 4 21 21 882 25x25 3 6 19 19 1083 25x25 4 8 17 17 1156 Max 25x25 5 10 15 15 1125 25x25 6 12 13 13 1014 25x25 7 14 11 11 847 25x25 8 16 9 9 648 25x25 9 18 7 7 441 25x25 10 20 5 5 250 25x25 11 22 3 3 99 25x25 12 24 1 1 12

25x25

4.1

8.2

16.8

16.8

1157.184

##### Max

25x25

4.2

8.4

16.6

16.6

1157.352

25x25

4.3

8.6

16.4

16.4

1156.528

25x25

4.4

8.8

16.2

16.2

1154.736

25x25

4.5

9

16

16

1152

25x25

4.6

9.2

15.8

15.8

1148.344

25x25

4.7

9.4

15.6

15.6

1143.792

25x25

4.8

9.6

15.4

15.4

1138.368

25x25

4.9

9.8

15.2

15.2

1132.096

25x25

5

10

15

15

1125

25x25

5.1

10.2

14.8

14.8

1117.104

25x25

5.2

10.4

14.6

14.6

1108.432

25x25

5.3

10.6

14.4

14.4

1099.008

25x25

5.4

10.8

14.2

14.2

1088.856

25x25

5.5

11

14

14

1078

25x25

5.6

11.2

13.8

13.8

1066.464

25x25

5.7

11.4

13.6

13.6

1054.272

25x25

5.8

11.6

13.4

13.4

1041.448

25x25

5.9

11.8

13.2

13.2

1028.016

 25x25 4.11 8.22 16.78 16.78 1157.25 25x25 4.12 8.24 16.76 16.76 1157.3 25x25 4.13 8.26 16.74 16.74 1157.34 25x25 4.14 8.28 16.72 16.72 1157.37 25x25 4.15 8.3 16.7 16.7 1157.39 25x25 4.16 8.32 16.68 16.68 1157.41 25x25 4.17 8.34 16.66 16.66 1157.41 Max 25x25 4.18 8.36 16.64 16.64 1157.4 25x25 4.19 8.38 16.62 16.62 1157.38
 25x25 4.161 8.322 16.678 16.678 1157.405801 25x25 4.162 8.324 16.676 16.676 1157.406318 25x25 4.163 8.326 16.674 16.674 1157.406735 25x25 4.164 8.328 16.672 16.672 1157.407052 25x25 4.165 8.33 16.67 16.67 1157.407269 25x25 4.166 8.332 16.668 16.668 1157.407385 Max 25x25 4.167 8.334 16.666 16.666 1157.407402 25x25 4.168 8.336 16.664 16.664 1157.407319

Conclusion

I worked out the proportion that needs to be cut off the box to give maximum volume this was 0.24445, which is very close to ¼.

Ð stands for delta.

Firstly we should consider a graph of y = x² as shown below.

The line through X and Y has almost the correct gradient.

Increase in y-coordinate from X to Y

Increase in x-coordinate from X to Y

You have to find an expression for , which represents
the gradient of the graph at the point X.

So

y = x²

y + Ðy = (x + Ðx)(x +Ðx)

Multiply out brackets.

y+ Ðy = x² + x Ðx + x Ðx + (Ðx) ²

y + Ðy = x² + 2xÐx + (Ðx)²

Now here the x² at the end is y in terms of x.

Ðy = x² + 2xÐx + (Ðx) ² - x²

Then you divide by Ðx, which gives you

Ðy    2xÐx – (Ðx) ²

Ðx =         Ðx

Ðy

Ðx   = 2x + Ðx

Now because delta (Ð) is so tiny that it is insignificant, we forget all about it, which leaves us with

Ðy/Ðx = 2x = 0

I can use calculus to help me complete my calculations to solve the problem, for proof through exhaustion.

I have drawn a graph showing the proportions that I have worked out. I can see that they tend towards ¼. This is the amount that should be cut off each corner to give the maximum volume possible.

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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