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### Exponential Decay

```Date: 06/09/2005 at 22:30:14
Subject: Exponential Decay Curve of a Radioactive Substance

In science we recently had a problem where we had to extend an
exponential decay curve of a net counting rate of a radioactive
substance by 1 day.  We were given the information:

Time (minutes)    Net counting rate (counts / minute)
0   (start)             171
20                       110
40                        70
60                        49
80                        31

When I extended the curve by one day I recieved 1.61 * 10^(-11) (which
is an estimate because the numbers that we were given would not always
turn out the same.)  My teacher insists that it would be zero.  I am
wondering who is correct.

I believe that I did my calculation correctly, but I am not sure if
the matter could reach that low a rate or not.  I don't think it can
reach 0 at all because there is always something left after each decay
cycle.

Could you please inform me whether my thinking is correct on this and
whether or not my answer is close to what the real one would be?

```

```
Date: 06/10/2005 at 23:22:33
From: Doctor Greenie
Subject: Re: Exponential Decay Curve of a Radioactive Substance

I didn't check your calculations, but who is "right" depends on your
interpretation.

In your last sentence, you ask whether or not your answer is close to
what the real one would be.  The answer is that it is close.

And your teacher's answer is also close.  After all, there is not a
lot of difference between "0" and "1.61 * 10^(-11)".

One thing that must be kept in mind in all of this is that rates of
4*10^10 atoms of a radioactive material with a half-life of 1 year,
then after a year you can be pretty confident that the number of atoms
remaining after a year is very close to 2*10^10.  But if you start
with 40 atoms of the material, then after 1 year you can't have much
confidence at all in the number of atoms remaining.

Since your measurements are "counts per minute", an answer of
1.61*10^(-11) means the radioactivity is probably not measurable, so
an answer of "0" is just as good.

I am reminded of an experience I had 30 years ago when I (as a math
student mildly interested in nuclear physics) took a nuclear
engineering course with a group of students specifically studying to
be engineers.  We had a problem on a test with a radioactive source
and an absorptive shield and were supposed to determine the strength
of the radiation on the other side of the shield.

Radiation strength on the other side of a shield, as a function of the
thickness of the shield, is an exponential decay function just like
radioactive decay.  When I put the numbers into the equation and did
some quick mental approximate calculations, I got a mathematical
result of something like e^(-60).  So without any further effort, I
wrote down the answer "0".  Several of the engineering students I
talked to after the test said they couldn't answer the question
because, whenever they put the equation into their calculators,
instead of getting an answer, their calculators kept showing
"0"....!!!

I hope some of this helps.  Please write back if you have any further
questions about any of this.

- Doctor Greenie, The Math Forum
http://mathforum.org/dr.math/
```
Associated Topics:
College Exponents
College Physics
High School Exponents
High School Physics/Chemistry

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