Carbon Dating

Date: 05/27/99 at 07:33:49
From: Vicky
Subject: Decay (Carbon Dating)

Bones A and B are x and y thousands years old respectively. Bone A 
contains three times as much Carbon-14 as bone B. What can you say 
about x and y?

Date: 05/27/99 at 10:24:29
From: Doctor Anthony
Subject: Re: Decay (Carbon Dating)

If r = ratio of carbon-14 to carbon-12 then, if we take r = 1 in 
living material (say branch of a tree) then with the half-life of 5730 
years we have:

    In 5730 years the value of r is down to 1/2

So we have the equation   r = (1/2)^(t/5730)  where t = time in years.

With this equation we can see that t = 0 gives r = 1

             "                     t = 5730 gives  r = 1/2

             "                     t = 11460 gives  r = 1/4

and so on.

So for bone A we have   r(A) = (1/2)^(x/5730)

and for bone B we have  r(B) = (1/2)^(y/5730)

and we are told         r(A) = 3*r(B)

              (1/2)^(x/5730) = 3*(1/2)^(y/3730)

     (1/2)^(x/5730 - y/5730) = 3          and taking logs

          (x-y)/5730 ln(1/2) = ln(3)

                  (x-y)/5730 = ln(3)/ln(1/2)

                  (x-y)/5730 = -1.584962

                       x - y = -9082

                           x = y - 9082

and so bone A is about 9000 years younger than bone B

- Doctor Anthony, The Math Forum
  http://mathforum.org/dr.math/