One of my students just can’t wrap his head around the power of exponents. Can you blame him? This week we learned that it would take just 27 folds of a piece of paper for the stack to reach the height of Mount Everest, and then just 15 more -a total of just 42 folds - to reach the moon. As we started the lesson, students guessed “one million” and “47 billion!” folds to reach the moon, so you can imagine the shock (and disbelief) in the actual number 42! Maybe the weirdest part is to think that it would take 41 of the folds to get just half-way to the moon and then just 1 more to make the second half of the journey.
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But how can this be? How is it possible that a thin sheet of paper easily ripped in half can reach the moon after a mere 42 folds? Well, let’s see….
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The simple Algebra 1 exponential growth formula is:
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As the thickness of a piece of paper is roughly 0.01 centimeter, we’d fill in our equation as:
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This gives us a very large number of centimeters: (43,980,465,111).
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Dividing this number by 100 will give us the equivalent number of meters: (439,804,651),
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and dividing by 1,000 will give us the equivalent number of kilometers: (439,804).
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For all us Americans stuck on the Imperial system, 439,804 kilometers is approximately 273,281 miles. The moon is, on average, 238,855 miles from Earth at any given time, so 42 folds of a piece of paper will actually get us PAST the moon!
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So how small would the surface area of the top paper on the stack be? How thin will be this paper tower to the moon?
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VERY thin!
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The exponential decay formula is almost exactly the same as the exponential growth formula except that there is a (1 – r) in place of the growth formula’s (1 + r). To write the equation for how thin this stack of paper to the moon will be, we have to think about a funny occurrence in the stock market…
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To go from 1 to 2 is a 100% increase: 100% of 1 is added to itself to get 2.
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But to get back to 1 is a different story: to go from 2 to 1 is a 50% decrease. Just 50% of 2 is removed to get to 1.
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So when your stock goes down 30% on Monday, it’s not back to where it was if it goes back up 30% on Tuesday. If your stock goes down 50% on Monday, it’s got to go up 100% on Tuesday to get back to where it was.
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Getting back on track (excuse me, not a fan of Wall Street), our decay equation would be written as:-
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This equation yields the incredibly small number: 6.37 x 10^-12 or .00000000000637 centimeters. P was set to 28 to because a 9.5×11 sheet of paper is about 28 centimeters long.
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We didn’t do this last part in class, which is a good thing because all of a sudden I’m having a hard time wrapping my head around exponents!
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