Carbon-14 dating?

Question

A tree killed by being buried under volcanic ash was found to have a ratio of carbon-14 atoms to carbon-12 atoms of 4.8x10^-14. How long ago did the eruption occur?

(The answer is 2.7x10^4, although I'm not sure how they got that answer... could anyone explain what I have to do?)

Answer 1

Carbon isotopes C-12 and C-13 are stable isotopes. However, C-14 is not, and its half-life is 5.70x10^3 years. When the true was alive, the ratio of C-14 to C-12 inside the body of the tree must be of the Natural abundance, that is 1 part per trillion: 10^-12. After the tree got buried under volcanic ash, there would be no flesh carbon supply to the tree any more, and hence the amount of C-14 would become less and less due to radioactive decay.
Let X years be the time after the eruption. We have:
4.8x10^-14/10^-12 = 0.048 = 0.5^(X/5.70x10^3)
X = 5.70x10^3*log(base2)(1/0.048)
= 2.5x10^4 (years)
2.5x10^4 years and 2.7x10^4 years are very well comparable, since we use different data sources.

My data sources:
http://en.wikipedia.org/wiki/Isotopes_of_carbon
http://en.wikipedia.org/wiki/Carbon-14