Paper and the Moon - posted April 12, 1999

A sheet of paper is .016 cm thick. Suppose that you tear this paper in half. Then you stack the two halves together and tear them in half. Then you take the four pieces, stack them, and rip them in half. If it were possible, you would continue this process of stacking the ripped pieces together and tearing them apart 60 times.

You need to answer the following four questions correctly to receive credit. All answers must be written in scientific notation.

1. How many pieces would the stack have?
   
   
2. How high would the stack reach?
   
   
3. Would it be taller than the highest building (not under construction)? How high is the tallest building?
   
   
4. Would it be higher than the moon at its greatest distance from Earth? What is this distance?

Be sure to show all your work and explain how you got your answers.

Need help in converting in the metric system? Try visiting the Learning Network's Weights & Measures [http://www.infoplease.com/ipa/A0001657.html].

Need to find information about the tallest building? Visit the World's Tallest Buildings [http://www.infoplease.com/ipa/A0001338.html].

Need to know the distance from the moon to the Earth? Check out The Moon [http://www.infoplease.com/ipa/A0004434.html].

Comments

I had several objectives when posing this week's problem. I wanted students to understand the exponential nature of the first problem. I wanted them to practice writing large numbers in scientific notation. I wanted to give them more experience in numerical conversions and comparisons. Finally, I wanted students to get information using the Internet.

Most errors occurred in the conversions. I was not specific about how students could make the conversions, and I received every possible form. Most students who got credit converted their answers to centimeters or some other metric form. This actually was the easiest way to express the answers.

Whether students picked the Sears Tower as the tallest building or the other Tower buildings in the world (some actually are no longer under construction) really did not matter. Since the stack of papers was higher than the distance to the moon, it was clearly taller than any mere building on earth.

Highlighted solutions:

If a piece of paper is torn in half and stacked and that new pile

torn and

stacked 60 times the resulting height of the pile would extend

approximately

1.8446X10^11km. This extends well past the moon that is only

approx. 4.066x10^5 km. So, yes, the stack is taller than either

Petronas Tower 1, or the Sears Tower.

2^60 will give me how many pieces of paper I would have at the end of

60 stack and tears. If each piece was .016 cm thick, which is

(1.6x10^-7km) then

(1.6x10^-7km)(2^60)=

(1.6x10^-7 km)(1.152921505x10^18 )=1.844674407x10^11 km



The distance from the Earth to the moon at it's farthest point is:

(252,710 miles)(1.609 kilometers/mile)= 406,610.39 kilometers

406,610.39 kilometers=4.0661039x10^5 km





Petronas Tower 1, Kuala Lumpur, Malaysia UC98  1,483 ft (I include

this building, because it was to be completed in 1998)

Sears Tower, Chicago 1974  1,450 ft



The closest the Moon can come to us (its perigee) is 221,463 miles;

the farthest it can go away (its apogee) is 252,710 miles.

1 mile (statute or  land)  5,280 feet =1.609 kilometers

1. This stack would have 2^60, or approximately 1.1529215 * 10^18

pieces.

2. This stack would reach up to 2^20 * 0.016 cm, or approximately

1.8446744 * 10^16 cm high.

3. Yes, the stack would reach higher than the tallest building. The

tallest building in the world is 4.62 * 10^4 cm.

4. Yes, the stack would reach farther than the moon. The moon at it's

perigee is approximately 3.5633396 * 10^10 cm from us. The moon at

it's apogee is approximately 4.0661039 * 10^10 cm from us.

First, each cut gives us 2 times the number of pieces, since we are

cutting them into two parts. So if we have 60 cuts, and there was only

one sheet when we started, there will be 2^60 sheets of small paper

when we finish the cutting. When we put this in on a calculator, we

find that this is approximately equal to 1.1529215 * 10^18 pieces.

Second, notice that each piece is the same thickness, which is 0.016

cm. Now, we will multiply that by the number of pieces of paper. So

the height of the stack is 0.016 * 2^60, which is approximately equal

to 1.8446744 * 10^16.

Third, we need to find out the highest building on earth. We find out

that that is 462 metres, which is equal to 46200 cm. So we can now

rewrite this in scientific notation, which is 4.62 * 10^4. Well, it is

quite obvious that the stack is much higher than this.

Fourth, we find that the moon at it's perigee is 221463 miles, and

it's apogee is 252710 miles. Now, note that we want this in

centimetres. Well, 1.609 kilometres is equal to one mile. Now, we can

multiply the perigee and apogee by this and we will get the number of

kilometres. So the perigee and apogee are approximately 356333.967 and

406610.39 cm from the earth respectively. Now, we need to multiply by

1000 to convert into metres, and multiply by 100 to convert metres

into centimetres. So the perigee and the apogee are 356333.967 * 1000

* 100 cm and 406610.39 * 1000 * 100 cm, respectively. This is

equivalent to 3.5633396 * 10^10 cm, and 4.0661039 * 10^10 cm. We can

now see that the stack is longer than this distance all the time, even

at the apogee.

1.152921504607e+18 pieces would be 1.844674407371e+14 meters high,

much higher than the tallest building not currently under

construction, the Petronas Towers in Kuala Lumpur, Malaysia which is

only 4.52e+2 meters tall and even higher than the furthest distance

from the earth to the moon,  4.06610390e+8 meters!

The general idea of the problem is to go in order, determining 1.

How many pieces there are 2.  How much distance would be created by

stacking these pieces on top of each other and numbers 3. and 4.

using the resources provided to determine if the height is < or >

than the tallest building not currently under contruction and < or >

the furthest distance from the surface of earth to the surface of the

moon.



1.  The number of pieces of paper multiplied by the width of each

piece of paper will approximate the height of the stack of paper.



-The number of pieces of paper follows tthe pattern 2^x;  x being the

number of times the paper is divided.  2^60= 1.152921504607e+18

pieces!



2.  The given width of these pieces of paper is .016 cm.



The resulting height of the pieces of paper stacked on top of each

other would be approximately (the number of pieces x the width)

1.152921504607e+18 x .016 cm =  1.844674407371e+16 cm   or

1.844674407371e+14 m



3.  Comparing this height with the height of the tallest building,

not under construction (which is done by looking at the provided

information found at the http://www.infoplease.com/ipa/A0001338.html

site and determining that it has already been completed '98) clearly

shows that the paper would indeed be taller than the highest building

ever constructed, the Petronas Towers in Kuala Lumpur, Malaysia which

are only 4.52e+2 meters tall!



4.  Even more amazing is the fact that the furthest distance from the

surface of the earth to the moon (approximately) is (according to the

http://www.infoplease.com/ipa/A0004434.html site) 252,710 miles or in

terms of meters 252,710 miles x 1609.0 meters/mile =  406,610,390

meters or 4.06610390e+8 m!

1.      The stack would have 1.15292  x 10 (18 power) pieces.

2.  The stack would be 1.84467  x 10 (11 power) kilometres high

3.  Yes, it would be much taller than Sears Tower (442 metres).

4.  Yes, it would be much higher than the moon at its furthest

        distance (406,610 km.)

1.  The number of pieces in 60 rips is calculated as 2 (power 60)

= 1.15292 x 10 (18 power).



2.  The height of the stack can be calculated by multiplying the

number of pieces by the thickness of the paper:

    .016 cm   or   .00016 metres   or   .00000016 kilometres



        1.15292 x 10(18 power)   x  .00000016 km.   =

         1.84467 x 10(11 power) km.



3.  The Sears Tower was given on the internet as 442  metres.



4.  The distance to the moon was calculated as 252,710 miles x

         1.609 km. per mile

     = 406,610 km   or 4.0661 x 10(5 power) km.



Bonus:

5.  At which rip does the stack height exceed the Sears Tower?

6.  At which rip does the stack height exceed the moon?

Answer:

5.  The number of pieces need to match the Sears Tower is

    442 m / .00016 metres = 2,762,500 pieces.

    Log (2,762,500) / Log (2) = 21.4

    The stack exceeds the Sears Tower on the 22nd rip.

    The stack goes from 2,097,152 pieces to 4,194,304 pieces.



6.  The number of pieces need to match the moon is

    406,610 km / .00000016 km = 2.54131 x 10(12 power) pieces.

    Log (2.54131 x 10(12 power)) / Log (2) = 41.2

    The stack exceeds the moon on the 42nd rip.

    The stack goes from 2.2 x 10(12 power) pieces to 4.4 x 10(12

        power) pieces.

The paper stack would contain 1.152921505 X 10^18 pieces.  The stack

would be 1.844674407 X 10^16 cm tall.  The highest building is The

Sears Tower which is 44196 cm in height; the stack is taller than the

world's tallest building.  The farthest distance the moon can reach

from the earth is 4.066973222 X 10^10 cm.  The stack is taller than

To answer the first question I first created a table, beginning with X

= the number of folds, and Y = the number of pieces in the stack.  The

table began



X  |  Y



0     1



1     2



2     4



3     8



and so on.  Examining the table it becomes evident that the growth is

clearly exponential.  The Y values double with each subsequent X

value.  This particular growth pattern is characteristic of the

equation Y = 2 ^ X.  That equation was entered into a graphing

calculator; my table was identical.  To make life easier, I used the

calculator to scroll down to the point where X = 60.  At this point

Y=1.152921505 X 10^18; thus the answer for question 1 was found.



       For question 2 I simply multiplied the number of pieces in the

stack by the thickness of one sheet, .016 cm.



So : (1.152921505 X 10^18) X  .016  =  1.844674407 X 10^16 cm = the

height of the stack.



       The height of the tallest building, 1450 ft, now needed to be

converted to centimeters.







1450 ft  X  ( 12 in / 1 ft ) X ( 2.54 cm / 1 in ) = 44196 cm.







The stack is higher than the world's tallest building.



         The moon's farthest distance from earth is 252720 miles.

This also needed to be converted into centimeters.







252710 mi  X  ( 5280 ft / 1 mi )  X  ( 12 in / 1 ft )  X  ( 2.54 cm /

1 in ) = 4.066973222 X 10 ^ 10 cm.



The stack is taller than the farthest distance the moon can reach from

t

1. The stack would have 1.1529*10^18 pieces of paper.

2. The stack would reach 1.8446*10^16cm.

3. Yes. The tallest building is 4.62*10^4cm.

4. Yes. The furthest distance to the moon is 4.0667*10^10.

To find how many pieces of paper there are in the stack, I did 2^60. I

did that because you are ripping the paper in half 60 times. To make

sure that I was doing the right thing I used a smaller number in place

of 60. I chose to rip a paper in half 3 times.



Rip    Pieces of paper

1 2

2 4

3 8



2^3 = 8



It worked. So I did 2^60 on my calculator and got 1.1529*10^18. Lucky

for me my calculator already did scientific notation.



To find out how high the stack would be I did 0.016*2^60. I did this

because I knew how many papers were in the stack and that each paper

is 0.016cm thick. So I multiplied 0.016 by 1.1529*10^18. I got

1.8446*10^16cm.



I found out that the tallest building is 462meters(m). 1m = 100cm So I

knew that the tallest building in centimeters is 46200. I got that by

multiplying 462 by 100. To put 46200 in scientific notation I took

away the two zeros on the end and made that 10^2. So I got this:

462*10^2. I knew this was not in scientific notation because in

scientific notation you need to multiply a number less than 10 by a

power of 10. So I put a decimal after the 4. Since I did that I knew I

had to multiply 4.62*10^2 by another 100, so I made 10^2 10^4. My

scientific notation is 4.62^10^4. I knew the stack of paper would be

higher because the number you square 10 by is higher in absolute

value. The number you square 10 by means you move the decimal point

that many places to the right.



The furthest distance to the moon is 252710miles(mi). I found out to

go from meters to miles you need to multiply the number of meters by

0.0006214. I needed to go from miles to meters. I knew that to go from

miles to meters you need to divide by 0.0006214. So I divided 252710mi

by 0.0006214 and got 406678468m. Then I multiplied 406678468m by 100,

on my calculator, because I needed to get the distance in centimeters.

I got 4.0667*10^10cm.

1)1.15*10^18pieces

2)1.84*10^16cm

3)yes,it would be higher. tallest building=4.62*10^4cm

4)yes,it would be higher. the distance=4.066*10^10cm

Make a list to find a relationship.



How many times you tear    1    2    3    4    5

  Height of the stock    .032 .064 .128 .256 .512

 # of pieces of paper      2    4    8   16   32



Let how many times you tear, X and let H represent the height of the

stock, P represent # of pieces of paper.



There's a relationship, H=.032*2^(X-1)  P=2^X

We tear the paper 60times, so let X=60 and solve the equation.



1)P=2^X                  2)H=.032*2^(X-1)

   =2^60                    =.032*2^(60-1)

   =1.15*10^18pieces        =.032*2^50

                            =.032*5.76*10^17

                            =1.84*10^16cm



3)The tallest building=460m

462m * 100cm = 4.62*10^4cm       Height of 1.84*10^16cm so the stock

        1m                       would be higher



4)The distance between the earth and the moon=252,710miles

252,710miles * 1609m * 100cm = 4.066*10^10cm

               1mile    1m

 The stock would be still higher than the furthest distance between

the earth and the moon.

1) There will be approximately 1.153 * 10^18 pieces of paper in the

stack.

2) The stack will be approximately 1.845 * 10^11 kilometers tall.

3) The stack will be taller than the Suyong Bay Tower, which is 462

meters tall.

4) The stack will reach farther than the moon's greatest distance from

Earth, which is approximately 4.069 * 10^5 kilometers.

1) Each time the piece(s) of paper are torn in half, their number

doubles. Starting with one piece and tearing it in half 60 times

yields 1 * 2^60 or about 1.153 * 10^18 pieces (1.153 quintillion

pieces).



2) The height of each piece of paper multiplied by the number of

pieces in the stack is (1.153 * 10^18) * 0.016 cm, which equals 1.845

* 10^16 cm. Divide by 100 to get meters, then divide by 1000 to get

kilometers, yielding 1.845 * 10^11 (184.5 billion) km.



3) The tallest building in the world is the Suyong Bay Tower, which is

462 meters tall. The height of the stack of paper is much more than

this.



4) At its farthest distance from the Earth, the Moon is 252,710 MILES

away. Multiplying by 1.61 gives kilometers, showing that the Moon is

about 4.069 * 10^5 (406.9 thousand) km away. The stack is even higher

than this!

1.  There would be 1.152921504607*10^18 pieces in the stack.

2.  The stack would reach 1.8 * 10^11 km. (two significant digits

    since that's how the thickness of the paper is given).

3.  It would be about 4.2 * 10^11 times taller than the Sears

Tower, the tallest building not under construction.  The Sears

Tower is 442 m (1,450 ft) tall.  The highest of all building

(including those under construction) is the Suyong Bay Tower 88,

in Pusan, S. Korea at 462 m (1,516 ft)

4. The stack would about 4.5 * 10^5 times taller than the

farthest distance to the moon.  The farthest distance to the

moon is given as 2.52716 * 10^*5 miles or 4.06707 * 10^5 km

l.  Each tear would be the next power of 2

    Before you tear:

            2^0 = 1 piece

    then each tearing doubles the number of pieces.

            2^1 = 2 pieces

            2^2 = 4  "

            2^3 = 8  "

            ....

           2^60 = 1.152921504607*10^18 pieces



2.  Multiply the number of pieces by the thickness of the paper

    to get:



    2^60 * 0.016 cm = 1.844674407371 * 10^16 cm

         1 m/100 cm = 1.844674407371 * 10^14 m

        1 km/1000 m = 1.844674407371 * 10^11 km



3.  The tallest building not under construction as given on

    http://www.infoplease.com/ipa/A0001338.html is the Sears

    Tower @  442 m (1450 ft).  The stack is  417,347,150,988.9

    (4.173471509889 * 10^11) times taller than the Sears Tower.



4.  The farthest distance of the moon is given as 252,716 miles

    252,716 mi * 1.60934 km/mi = 406,707 km (4.06707 * 10^5 km)



  1.844674407371 * 10^11 km

    --------------------------  =  4.535634762547 * 10^5 times

    4.06707     *    10^5  km      farther than the moon



--------



For other comparisions I calculated that light would take about

7.1 days to travel from the top and that this stack is about 16

times the diameter of the solar system.

The stack would have  1.152921505 E18 pieces of paper.

The stack would reach 1.844674407 E16 cm high.

Yes, it would be taller than the highest building, which is 4.42 E4 cm

high.

Yes, it would be higher than the furthest distance from the moon to

Earth, which is 4.06686203 E10.

   First, to understand what the problem was asking, I started to tear

up a piece of paper.  When I tore the piece of paper 1 time, I got 2

pieces.  When I tore the pieces of paper 2 times, I got 4 pieces.

When I tore the pieces of paper 3 times, I got 8 pieces.  After this,

I tried to figure out a pattern that was going on.  I realized that 2

raised to the number of times the paper was torn, would equal the

answer.  For example, 2 raised to the first equals 2.  2 raised to the

second equals 4. 2 raised to the 3 equals 8, and so on....  Then to

answer the first question, I tried 2 to the 60th, which equals

1.152921505 E18.

   For the next question, I multiplied the first answer by .016 cm

because that is how high each sheet of paper is.  The next answer I

got was 1.844674407 E16 cm.

   Then I went to the website suggested to find out how high the

tallest building is.  I found out that the Sears Tower is the tallest

building not under construction.  It is 442 meters tall.  I multiplied

442 by 100, to convert meters into centimeters.  So, in scientific

notation, this amount is equal to 4.42 E4.  Since, 1.844674407 E16 is

greater than 4.42 E4, the stack would be taller than Sears Tower.

   Then I went to the next website suggested to find out far the moon

is at its furthest distance from earth.  It said that the moon was

252,710 miles from earth.  Then I went to the other website listed,

that gave conversion factors.  I changed 252,710 into kilometers,

which was 406686.203.  Then I converted this answer into centimeters

by multiplying it by 100,000.  Then I changed the answer into

scientific notation, which gave me 4.06686203 E10.  Since 4.06686203

E10 is less than 1.844674407 E16, the stack of paper would be taller

than the furthest distance from Earth to the moon.

55 students received credit this week.

Madeel Abdullah, age 15 - Marple Newtown Senior High School, Newtown Square, PA
N Alberti, age 13 - Far Hills Country Day School, Far Hills, NJ
Martin Angert, age 14 - Holland Junior High School, Holland, PA
Mike Bockus, age 16 - Wilburton High School, Wilburton, OK
Jill Burkhart, age 16 - Shaler Area High School, Pittsburgh, PA
Pat Caffrey, age 14 - West Morris Central High School, Chester, NJ
Christopher Cellini, age 15 - Notre Dame High School, West Haven, CT
Katie Clark, age 16 - Shaler Area High School, Pittsburgh, PA
Monica-Ramona Costache, age 14 - Mihai Viteazul High School, Bucharest, Romania
Lee Crandell, age 25+ - none, Duarte, CA
B.R Cutler, age 14 - Roosevelt High School, Minneapolis, MN
Nicole Czake, age 16 - Shaler Area High School, Pittsburgh, PA
Brendan Dilloughery, age 16 - Soquel High School, Soquel, CA
Brenna Dolphin, age 14 - Wilmington Friends, Wilmington, DE
Jerrod Early, age 16 - Shaler Area High School, Pittsburgh, PA
Jared Elizares, age 14 - Amador Valley High School, Pleasanton, CA
Jenn Elliott, age 16 - Marple Newtown Senior High School, Newtown Square, PA
Ariel Franks, age 11 - Forsyth School, St. Louis, MO
Mariko Furukawa, age 16 - Shaler Area High School, Pittsburgh, PA
Lisa Gardner, age 8 - Boyette Springs Elementary School, Riverview, FL
Kristen Geubtner, age 15 - Shaler Area High School, Pittsburgh, PA
Jen Gordon, age 15 - Marple Newtown Senior High School, Newtown Square, PA
Chris Hardesty, age 12 - Strickland Middle School, Denton, TX
Natalie Hess, average age 14 - Wilmington Friends, Wilmington, DE
Andrea Hsu, age 13 - William Annin Middle School, Basking Ridge, NJ
Junior High Project Challenge Class, average age 13 - St. Angelas Church School, Cleveland, OH
Dave Kennedy, age 25+ - none, Lethbridge, Canada
Richard Kilcoyne, age 14 - West Essex Jr. High, North Caldwell, NJ
Chrissy Laverdier, age 19 - Arizona State University, Tempe, AZ
Chen-Wen Lee, age 12 - Strickland Middle School, Denton, TX
Mark Lessmueller, age 14 - none, Sacramento, CA
Russ Lutz, age 16 - Shaler Area High School, Pittsburgh, PA
Jon Mankovich, age 16 - Wallkill Valley High School, Hamburg, NJ
Math Club, average age 14 - Sweet Home, Amherst, NY
Martin McNicoll, age 19 - Strathclyde University, Glasgow, Scotland, UK
Erin Meyer, average age 15 - Shaler Area High School, Pittsburgh, PA
Jeffrey Mo, age 9 - University Elementary School, Calgary, Alberta, Canada
Preeti Nalavade, age 12 - Wissahickon Middle School, Ambler, PA
Ryan Newman, age 16 - Shaler Area High School, Pittsburgh, PA
Joseph O'Connor, age 25+ - none, Pleasant Grove, UT
Joe Pacold, age 13 - Calvert Homeschooling, Baltimore, IL
Angie Palmer, age 19 - Mesa State College, Grand Junction, CO
Faye Paul, age 14 - Wilmington Friends, Wilmington, DE
Laura Pelat, age 15 - Shaler Area High School, Pittsburgh, PA
Michael Pizer, age 10 - University School of Milwaukee, Milwaukee, WI
Olga Rodina, age 14 - Winman Junior High School, Warwick, RI
Kristen Rogerson, age 17 - Fr.Leo J.Austin Catholic Secondary School, Whitby, Canada
Laura Severance, age 14 - Wilmington Friends, Wilmington, DE
Brett Shiring, age 16 - Shaler Area High School, Pittsburgh, PA
Aaron Sichmeller, age 15 - Watertown Senior High School, Watertown, SD
Adam Starling, age 17 - P.S.C.I, Orillia, Canada
Nathan Strauss, age 11 - Forsyth School, St. Louis, MO
Gerjanne Vlasveld, age 14 - Willem van Oranje College, Waalwijk, Netherlands
James Western, age 15 - Wallkill Valley High School, Hamburg, NJ
Doug Wolfe, age 16 - Shaler Area High School, Pittsburgh, PA

View most of the solutions submitted by the students above