THE SOLUTION TO Weekly Challenge 3: What Can You Do With a Humongous Piece of Xerox Paper?
Posted by DrJeff on June 23rd, 2009
Filed under 1.1. Dr. Jeff's Weekly Challenge, 1.2. Solutions to Weekly Challenges, 1.5. Dr. Jeff's Jeffisms, 3. Science Education, 4. The Earth, 5. Space Science, 5.1. Our Solar System, 5.1.1. The Sun, 5.1.2. The Moon, 5.4. Milky Way Galaxy, 5.5. Other Galaxies, 5.6. The Universe
Copyright 2009 | About this blog
Read Original Challenge HERE.
This post is a Dr. Jeff’s Weekly Challenge and a Dr. Jeff’s Jeffism.
Last week on BotU, your challenge was to take an imaginary, truly humongous piece of xerox paper—but with normal xerox paper thickness—and figure out how many times you’d need to fold it in half so the folded thickness is the height of you, the Statue of Liberty, the Empire State Building, and Mount Everest. For those that really wanted to challenge themselves, I invited you to keep folding so it would be thick enough to reach the Moon, the Sun, the nearest star, and beyond.
How’d you do?
BUT WAIT! If you haven’t yet read Weekly Challenge 3, DON’T LOOK AT THE SOLUTION HERE JUST YET! First read Weekly Challenge 3, or I’ll take back my paper.
First, a word from our sponsor—
You Want Me To Do What With a Bathroom Scale?
Weekly Challenge 4 to be posted Monday, June 29, 2009
Other Posts coming soon:
A Voyage in Corpus Christi
Twinkle Twinkle Little Star, History Tells How Far You Are
Lessons of Earth
MESSENGER: Target Mercury
And now the answers—
Remember that I began Weekly Challenge 3 by asking you to imagine a humongous piece of paper that, when standing in the middle of it, seemingly extends to the horizon in all directions. This is clearly NOT a real piece of paper. What I posed is called a Thought Experiment—an experiment done on the landscape of your mind. It’s a flavor of experiment that has been done by the likes of Einstein to revolutionize our understanding of space and time. A Thought Experiment often poses a problem that cannot be addressed with a real experiment because doing so is either impossible (as in this case) or clearly beyond current technology, or for that matter beyond any interest in doing it in the real world (e.g., where would you come out if you dug a hole right through the center of the Earth?) And this is precisely why it’s such a powerful type of experiment. It uses critical thinking coupled with what you think you know about the world, to hopefully produce a real conclusion—which often leads to a change in perception.
A note about trying to do this in the real world: You CAN’T actually fold a sheet of paper more than a few times because it quickly gets too small and all the paper is taken up in the curvature of the folds. Try it. If you want to increase the number of folds, you need a larger sheet of paper. But just a few more folds quickly requires a sheet that’s longer than your street, your city, your nation, and, in fact far longer than planet Earth is wide.
This problem was pointed out in the great comment by Maria Miller (see Weekly Challenger 3 page.) She has a link to a story that made news a few years ago. In 2001, Britney Gallivan a high school student set out to understand the limitations imposed by folding as part of an extra credit problem in math class. She even came up with a limiting equation that defines the minimum size of the paper sheet you’d need in order to fold it a specific number of times. Here are more links to this very cool story about curiosity and drive: 1, 2, 3.
SO … IMAGINE a truly humongous sheet of paper—as long and wide as you need, and let’s explore what would happen to its thickness as you keep folding it.
The answers below are based on a very simple rule. You start with a single sheet that has the thickness of a regular sheet of xerox paper (0.1 mm for those that want to know), and every time you fold the sheet in half:
you double the thickness of the paper
No big thing right? You’d think that you’d need A LOT of folds to get a thickness equal to these large distances. Here you go—
1. How many folds would you need so the thickness is as tall as a ream of xerox
paper (500 sheets)?
just fold it in half 9 times and thickness is 512 sheets
2. How many folds so the thickness is as tall as you?
just fold it in half 14 times and thickness is 5 feet 5 inches (166.5 cm)
3. How many folds so the thickness is as tall as:
the Statue of Liberty?
Height is 305 feet (93 m)
just fold it in half 20 times and thickness is 350 feet (106 m)
Height is 1,472 feet (448.7 meters)
just fold it in half 22 times and thickness is slightly short at 1,398 feet (426 m)
The summit is at 29,029 feet or 5.5 miles (8.8 km) altitude
just fold it in half 27 times and thickness is 8.5 miles (13.6 km)
4. How many folds so the thickness is equal to:
the distance from Earth to the Moon?
average distance: 238,900 miles (384,400 km)
just fold it in half 42 times and thickness is 277,650 miles (446,840 km)
the distance from Earth to the Sun?
average Earth-Sun distance: 93,000,000 miles (149,600,000 km)
just fold it in half 51 times and thickness is 142,159,000 miles (228,780,000 km)
the distance to the nearest star Proxima Centauri?
distance of 4.2 light years
just fold it in half 69 times and thickness is 6.3 light years
the width of our Milky Way Galaxy?
diameter: 100,000 light years
just fold it in half 83 times and thickness is 104,000 light years
the distance to the Andromeda Galaxy—nearest large galaxy to the Milky Way?
distance: 2.5 million light years
just fold it in half 88 times and thickness is 3.3 million light years
the distance from Earth to the edge of the observable universe?
distance: 78 billion light years
just fold it in half 103 times and thickness is 109 billion light years
YOU DON’T BELIEVE ME? I KNEW you were going to say that. So I created two Tables that show you how I got these answers really easily. One Table is in English units (inches, feet, miles), and the other Table is in Metric units (cm, km). Choose your system of units and be amazed!
Teachers:
Here’s a way to really do the challenge in the classroom by removing the pesky folding issue. Remember that the basic rule is each fold doubles the thickness of the paper. So instead of folding, just create a pile of paper using this rule, so you’re ‘simulating’ the effect that folding has on the pile’s thickness.
Remember the cartons of xerox paper we used for the post A Day in the Life of the Earth? Go get them out again!! I hope you know the person in charge of the copy room, or you’ll be banned.
Start by placing a single sheet of xerox paper on the floor. Then ask students in your class, one by one, to simulate each fold by doubling the number of sheets in the pile. Here’s the way it ‘unfolds’ (no pun intended)—
• First student sees 1 sheet on the floor, so places 1 new sheet on top: pile
thickness is now 2 sheets; they’ve simulated the effect of fold 1
• Second student sees 2 sheets in the pile so places 2 new sheets on top:
thickness is now 4 sheets; they’ve simulated the effect of fold 2
• Third student sees 4 sheets in the pile so places 4 new sheets on top:
thickness is now 8 sheets; they’ve simulated the effect of fold 3
Have them keep going and see how far they get. You already have detailed notes on how it will turn out. It’s my write-up in the two Tables, one for English units and the other for Metric units.
Important note: once you get up to 512 sheets, you can start using whole reams of 500 sheets (so again no need to open more than 1 ream of paper.)
Another note about how to start the lesson: pose the original challenge to the class, and have them see how many times they can fold a sheet of paper. After they quickly see the insurmountable hurdle that the folding requirement imposes, see if they can come up with this way to do the experiment by ‘simulating’ each fold.
Photo credit: NASA and STScI
One Response to “THE SOLUTION TO Weekly Challenge 3: What Can You Do With a Humongous Piece of Xerox Paper?”
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Rick Regan Says:
June 27th, 2009 at 5:33 pmMy favorite illustration of the powers of two is the one that goes like this: if I give you 1 cent today, then 2 cents tomorrow, then 4 cents the next day, etc. for 30 days, how much money will you have? $10,737,418.23!
Not only does it illustrate the power of doubling, but it involves the sum of a geometric series (the sum from i=0 to 29 of 2^i = 2^30-1). It also avoids the practical troubles of folding, or of putting grains of rice on a chessboard (another popular illustration).
Of course you could skip the story, and go right to a table of powers of two (e.g. http://www.exploringbinary.com/a-table-of-nonnegative-powers-of-two/ ).
