If the half life of C-14 is 5730 years, what is the radio-carbon estimate of the age of a fragment that has 21% the C-14 of the atmosphere it was in?

Radio carbon dating is based on the fact that C-14 changes
to N-14 due to beta decay. A live organism has the same percentage of C-14 as that in
the atmosphere due to an exchange of carbon when it consumes food and expels CO2. After
the organism dies, the accumulated C-14 reduces as it decays and there is no
replacement.

By measuring the percentage of C-14 and
comparing it to the percentage in the atmosphere it is possible to find the approximate
age of an organism.

The amount of C-14 becomes half in 5730
years which is the half life of C-14.

Let the age of the
fragment be N years. The amount of C-14 left is equal to (1/2)^(N/5730). As the fragment
has 21% of the C-14 as that in the atmosphere.

21/100 =
(1/2)^(N/5730)

=> N/5730 = log(0.21)/
log(0.5)

=> N = 5730* log(0.21) /
log(0.5)

=> N =
5730*2.2515

=> N =
12901

The fragment is 12901 years
old.