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Technology Stocks : Data Race (NASDAQ: RACE) NEWS! 2 voice/data/fax: ONE LINE!
RACE 391.90+2.4%Nov 28 9:30 AM EST
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To: R. Allan Choiniere who wrote (25942)11/22/1997 3:48:00 AM
From: Kashish King  Read Replies (1) of 33268
 
There's this pond about the size of two football fields with some duckweed on it. The duckweed doubles every day. Nine days later the entire pond is covered in slimy green duckweed. On what day was the pond half filled with duckweed?

My intial reaction was Rastafarian until I realized that the answer was far simpler: If it doubles every day and if filled the pond in nine days it must have been half of that on the same day, by your stipulation. How easy can you get?

Proof:

Growth of the plant material is proportional to the amount present at any given time and it can be shown, rather easily, that this corresonds to the differential equation A( t ) = A * ( e ^ ( k * t ) ) where A is the amount, t is time, e is the magic number e, and k is some constant of the aforementioned proportionality.

The implication was that there was half as much at some time in the past as there is now that the pond is full . We can plug those conditions into our equation and discover what that constant of proportionality is for duck weed based on the assumption that we have an amount A after 9 days:

= > 2 * A = A * ( e ^ ( k * 9 ) )
=> e ^ ( k * 9 ) = 2
=> k * 9 = ln( 2 )
=> k = ln( 2 ) / 9
=> k = .077016

Now we have the all important constant of proportionality and, by extension, a general solution:

A( t ) = A * ( e ^ ( .077016 * t ) )

Given a general solution, I can now give you a specific answer for when the amount was half of the current amount A:

=> 1/2 * A = A * ( e ^ ( .077016 * t ) )
=> 1/ 2 = e ^ ( .077016 * t )
=> ln( 1/2 ) = .077016 * t
=> t = - 9

The reason we get -9 and not 9 should be self-evident.
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