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Title
Paper Folding

Problem Statement
If a piece of paper could be folded in half fifty times, what would be the thickness of the folded paper? A ream (500 sheets) of paper is 2 inches thick.

Problem setup

This problem requires readers to fold a piece of paper 50 times.  After folding the paper fifty times, we must determine how thickness of that sheet of paper.

 

Plans to Solve/Investigate the Problem

My initial plan to solve the problem was to try to fold the paper a couple of times to see if a pattern existed between the number of folds and the thickness.

 

Investigation/Exploration of the Problem

Deciding that it was almost impossible to fold a piece of paper fifty times, I came up with an exponential formula for determining the number of sheets of paper based on the number of folds.  The following spreadsheet was created, including the formulas for each category. 

number of folds

number of sheets

reams

inches

feet

yards

miles

 

2 (folds) ^ number of folds

number of sheets/500 sheets in a ream

 

 

 

 

0

1

0.002

0.004

0.000333333

0.000111111

6.31313E-08

1

2

0.004

0.008

0.000666667

0.000222222

1.26263E-07

2

4

0.008

0.016

0.001333333

0.000444444

2.52525E-07

3

8

0.016

0.032

0.002666667

0.000888889

5.05051E-07

4

16

0.032

0.064

0.005333333

0.001777778

1.0101E-06

5

32

0.064

0.128

0.010666667

0.003555556

2.0202E-06

6

64

0.128

0.256

0.021333333

0.007111111

4.0404E-06

7

128

0.256

0.512

0.042666667

0.014222222

8.08081E-06

8

256

0.512

1.024

0.085333333

0.028444444

1.61616E-05

9

512

1.024

2.048

0.170666667

0.056888889

3.23232E-05

10

1024

2.048

4.096

0.341333333

0.113777778

6.46465E-05

11

2048

4.096

8.192

0.682666667

0.227555556

0.000129293

12

4096

8.192

16.384

1.365333333

0.455111111

0.000258586

13

8192

16.384

32.768

2.730666667

0.910222222

0.000517172

14

16384

32.768

65.536

5.461333333

1.820444444

0.001034343

15

32768

65.536

131.072

10.92266667

3.640888889

0.002068687

16

65536

131.072

262.144

21.84533333

7.281777778

0.004137374

17

131072

262.144

524.288

43.69066667

14.56355556

0.008274747

18

262144

524.288

1048.576

87.38133333

29.12711111

0.016549495

19

524288

1048.576

2097.152

174.7626667

58.25422222

0.03309899

20

1048576

2097.152

4194.304

349.5253333

116.5084444

0.06619798

21

2097152

4194.304

8388.608

699.0506667

233.0168889

0.13239596

22

4194304

8388.608

16777.216

1398.101333

466.0337778

0.264791919

23

8388608

16777.216

33554.432

2796.202667

932.0675556

0.529583838

24

16777216

33554.432

67108.864

5592.405333

1864.135111

1.059167677

25

33554432

67108.864

134217.728

11184.81067

3728.270222

2.118335354

26

67108864

134217.728

268435.456

22369.62133

7456.540444

4.236670707

27

134217728

268435.456

536870.912

44739.24267

14913.08089

8.473341414

28

268435456

536870.912

1073741.824

89478.48533

29826.16178

16.94668283

29

536870912

1073741.824

2147483.648

178956.9707

59652.32356

33.89336566

30

1073741824

2147483.648

4294967.296

357913.9413

119304.6471

67.78673131

31

2147483648

4294967.296

8589934.592

715827.8827

238609.2942

135.5734626

32

4294967296

8589934.592

17179869.18

1431655.765

477218.5884

271.1469253

33

8589934592

17179869.18

34359738.37

2863311.531

954437.1769

542.2938505

34

17179869184

34359738.37

68719476.74

5726623.061

1908874.354

1084.587701

35

34359738368

68719476.74

137438953.5

11453246.12

3817748.708

2169.175402

36

68719476736

137438953.5

274877906.9

22906492.25

7635497.415

4338.350804

37

1.37439E+11

274877906.9

549755813.9

45812984.49

15270994.83

8676.701608

38

2.74878E+11

549755813.9

1099511628

91625968.98

30541989.66

17353.40322

39

5.49756E+11

1099511628

2199023256

183251938

61083979.32

34706.80643

40

1.09951E+12

2199023256

4398046511

366503875.9

122167958.6

69413.61286

41

2.19902E+12

4398046511

8796093022

733007751.9

244335917.3

138827.2257

42

4.39805E+12

8796093022

17592186044

1466015504

488671834.6

277654.4515

43

8.79609E+12

17592186044

35184372089

2932031007

977343669.1

555308.9029

44

1.75922E+13

35184372089

70368744178

5864062015

1954687338

1110617.806

45

3.51844E+13

70368744178

1.40737E+11

11728124030

3909374677

2221235.612

46

7.03687E+13

1.40737E+11

2.81475E+11

23456248059

7818749353

4442471.223

47

1.40737E+14

2.81475E+11

5.6295E+11

46912496118

15637498706

8884942.447

48

2.81475E+14

5.6295E+11

1.1259E+12

93824992237

31274997412

17769884.89

49

5.6295E+14

1.1259E+12

2.2518E+12

1.8765E+11

62549994825

35539769.79

50

1.1259E+15

2.2518E+12

4.5036E+12

3.753E+11

1.251E+11

71079539.57

The last couple of columns are metric conversions from inches to miles.  I kept converting in order to create a whole number that did not have scientific notation.  The number highlighted in yellow would indicate how many miles one sheet of paper would be if folded fifty times. 

 

Extensions of the Problem

One of the extensions discussed was how far this sheet of paper would be relative to the sun. 

Author & Contact
Mesha Rainey
M Rainey


Link(s) to resources, references, lesson plans, and/or other materials
None

 

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