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## #1 2005-12-07 04:32:32

NewL
Novice

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### Exponents... Simple right?

Objective -

understand the concept of exponential growth...

Alejandro is making ballots for an election.  He cuts a single piece of paper in in half making 1 into 2...
He stacks the sheets and cuts them again making 2 into 4 stacks and cuts again making 8 and so on.

2 x 2 x 2 x 2 x 2 = 32     2^5    OR  "2 TO THE 5TH POWER"

A. Cut a sheet of paper as Alejandro did, and count the ballots after each cut.
Make a table to show the number of ballots after I cut, 2 cuts, 3 cuts, and so on.

1st cut = 2 ballots
2nd cut = 4 ballots
3rd cut = 8 ballots

B. Look for a pattern in the way the number of ballots changes with each cut.
Use your observations to extend your table to show the number of ballots
for up to 10 cuts.

1st cut = 2 ballots
2nd cut = 4 ballots
3rd cut = 8 ballots
4th cut = 16 ballots
5th cut = 32 ballots
6th cut = 64 ballots
7th cut = 128 ballots... and so on

C.  If Alejandro made 20 cuts, how many ballots would he have? How many
ballots would he have if he made 30 cuts?

20 cuts = 1048576 ballots
30 cuts =  1073741824 ballots

... so far so good

here's where I need help,  I have spoken with the teacher and gotten the answers... I'd just like to see if others get the same answers.  Please provide your answers for the next two questions.

D. A stack of 250 sheets of the paper Alejandro is
using is 1 inch high. How high would a stack
of ballots be after 20 cuts? After 30 cuts?

E. How many cuts would Alejandro need to make
to have a stack of ballots 1 foot high?

Thanks Chuck

## #2 2005-12-07 05:19:24

mathsyperson
Moderator

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### Re: Exponents... Simple right?

D. You are told that 250 sheets of paper make 1 inch, so to get the height, just divide the amount of ballots by 250 and that is the amount of inches.

20 cuts: 1,048,576 ÷ 250 = 4,194 inches to the nearest inch. (≈117 yards)

30 cuts: 1,073,741,824 ÷ 250 = 4,294,967 inches to the nearest inch. (≈68 miles)

E. A stack of ballots 1 foot high would need 250*12 = 3,000 ballot papers.
The smallest amount of cuts Alejandro would need is 12, because 2^11 = 2,048 and 2^12 = 4,096.

Why did the vector cross the road?
It wanted to be normal.

## #3 2005-12-07 09:19:53

MathsIsFun

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### Re: Exponents... Simple right?

What a neat little exercise!

Ummm ... if you started with A4 sheets, would there be enough room to cast your vote after 20 cuts? 30 cuts?

"The physicists defer only to mathematicians, and the mathematicians defer only to God ..."  - Leon M. Lederman

## #4 2005-12-07 10:44:39

mathsyperson
Moderator

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### Re: Exponents... Simple right?

Hmm. A4 paper is 21x29.7 cm, so it has an area of 623.7cm²

After 20 cuts, this would be divided by 2^20, making the area of each ballot ≈ 0.06 mm².
You could, but you'd have to tell a precision computer to do it for you.

Why did the vector cross the road?
It wanted to be normal.

## #5 2005-12-07 10:51:52

irspow
Power Member

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### Re: Exponents... Simple right?

After only 20 cuts you would be left with a ballot .2900390625mm high by .205078125mm wide!! (That's 1/87 X 1/125 inches)  Since this far less than the normal diameter of mechanical pencil led I doubt that anyone could vote with it because even a "dot" would not fit on the ballot.

After 30 cuts it gets even more ridiculous.  The piece of paper (?) would be only 41 nanometers wide by 58 nanometers high.  That means you could place approximately 33000 of these ballots atop a human hair!!!