A living specimen in equilibrium with the atmosphere contains one atom of 14C (h

Question

A living specimen in equilibrium with the atmosphere contains one atom of 14C (half-life = 5730 yr) for every 7.70 *10^11 stable carbon atoms. A bone fragment contains 1.00*10^21 carbon nuclei. When the sample is placed inside a shielded beta counter with 75.0% counting efficiency, 82,000 counts are accumulated in one year. Assuming that the cosmic-ray flux and the Earth's atmosphere have not changed appreciably since the sample was formed, find the age of the sample. I know the answer is 2993 years, but don't know why. Please explain! THANK YOU!

Explanation / Answer

Here, number of original atoms = 7.70 *1011 * e0.693

                                                               = 15.4 *1011 atoms

=> the age of the sample = 5730 * 0.75 * ln[(15.4 *1011)/(7.70 *1011)]

                                          =    2993 years

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