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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

...read more.

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

1999.404

30x30

5.2

10.4

19.6

19.6

1997.632

30x30

5.3

10.6

19.4

19.4

1994.708

30x30

5.4

10.8

19.2

19.2

1990.656

30x30

5.5

11

19

19

1985.5

30x30

5.6

11.2

18.8

18.8

1979.264

30x30

5.7

11.4

18.6

18.6

1971.972

30x30

5.8

11.6

18.4

18.4

1963.648

30x30

5.9

11.8

18.2

18.2

1954.316

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

1157.246124

25x25

4.12

8.24

16.76

16.76

1157.298112

25x25

4.13

8.26

16.74

16.74

1157.339988

25x25

4.14

8.28

16.72

16.72

1157.371776

25x25

4.15

8.3

16.7

16.7

1157.3935

25x25

4.16

8.32

16.68

16.68

1157.405184

25x25

4.17

8.34

16.66

16.66

1157.406852

Max

25x25

4.18

8.36

16.64

16.64

1157.398528

25x25

4.19

8.38

16.62

16.62

1157.380236

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

...read more.

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.
It’s gradient is

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

Add like terms together.

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.

...read more.

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