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Exponential Decay

Date: 05/02/97 at 23:55:27
From: Tequilla
Subject: Logarithms and exponential growth/decay

Health officials found traces of Radium F beneath the local library.  
After 69 days, they observed that a certain amount of the substance 
had decayed to 1/sqrt(2) of its original mass.  Determine the 
half-life of Radium F.

I can't figure out what the original mass is, so I can't fit the 
information into the equation. 

Please help!

Date: 05/03/97 at 06:29:02
From: Doctor Mitteldorf
Subject: Re: logarithms and exponential growth/decay

Dear Tequilla,

There is a mistake in this question, which is part of the reason it is 
hard for you to understand.  It is not the MASS of the Radium F that 
is measured now and after 69 days; it is the RADIOACTIVITY of the 
sample including Radium F.  In other words, you put a Geiger counter 
next to the stuff on day 1 and you count 100 clicks in a minute.  Then 
you do the same thing on day 70 (69 days later) and you hear only 71 
clicks in a minute.  This means that there's .71 as much radioactive 
material in the sample as there was before.  You can't do it with the 
mass because each atom of Radium turns into an atom of something else 
that weighs almost as much and the problem doesn't tell you how much 
that something else weighs.

Now, once you understand this, the next problem is to understand how
exponential decay works.  You're used to change that is LINEAR, where
the change ADDS or SUBTRACTS THE SAME AMOUNT every day.  For example, 
there may be 100 counts per minute the first day, 99 the second day, 
98 the third day, each day subtracting exactly one count.  But 
radioactivity doesn't work this way.  Instead, the number of counts is 
MULTIPLIED by the same fraction each day.  This seems very odd until
you get used to it.  The only way to get used to it is to try lots of 
examples and notice what is happening.

This isn't exactly the answer to the question in your book, but it 
will help you understand.  Try writing down the numbers from the 
example above: 100,  99,  98,  97,  96,  ... all the way down to 0 on 
the 101st day.

Now let's suppose instead that each day the amount of radioactivity 
gets MULTIPLIED by 0.99.  Then the first day is 100, the second day is 
99 (similar to before).  But the third day is 99 * 0.99 = 98.01 (just 
a tiny bit more than before). The third day, it is 98.01*0.99 = 
97.0299.  The difference is a little greater now.

Use a calculator to write down the numbers for the first hundred days, 
then make a graph of the first set of numbers (where you subtract 1 
each time) and the second set (where you multiply by 0.99 each time).  
After you've done this, you will be on your way to understanding how 
exponential decay works.

Please write back after you do this, and we can take the next step 

-Doctor Mitteldorf,  The Math Forum
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Associated Topics:
High School Exponents

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