Radioactive Decay of an Isotope

Date: 01/05/98 at 23:43:57
From: Alf Butler
Subject: Radioactive decay of an isotope

Hi! I was working on a question for Precalculus over the decay of 
an the 226Ra isotope. I know it involves the use of logarithms 
(actually it can be an exponential function ae^x), but I'm having  
a little trouble fitting my information to my equation.  

If I have the half-life in years (1620) and the initial quantity (10g) 
then how do I find the amount of the isotope after 1000 years?  
Shouldn't I use the exponential decay model (y=ae^(-bx); b is greater 
than 0)?  Hope you can help!  Thanks so much.

Date: 01/06/98 at 16:26:18
From: Doctor Jennifer
Subject: Re: Radioactive decay of an isotope

Hi Alf!

First what you should do is make a chart with all the variables and 
what they stand for. That way it isn't so confusing.

  y - amount of 226 Ra
  a - initial amount of 226 Ra
  b - constant specific to 226 Ra
  x - time elapsed

Now you need to figure out what you know and what you don't.

  y - ??  (that's your question)
  a - 10 g
  b - ??  
  x - 1000 years

You now have 2 unknowns. You need to look at what other information 
was given. You have a half-life of 1620 years. How is that helpful?  
If you fill out the chart again you only have one unknown.

  y - 1/2 (see reason under a)
  a - 1 (Assume one.  Then half of one is 1/2)
  b - ??
  x - 1620 years

Now if you plug all these values in the equation y = ae^(-bx), you can 
find your constant for b. Then go back to the original equation and 
your values should plug in perfectly.

If you have any more problems don't hesitate to ask.  We're here to 
help.

-Doctor Jennifer,  The Math Forum
 Check out our web site!  http://mathforum.org/dr.math/