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The halflife of carbon 14 is 5730 ± 30 years, and the method of dating lies in trying to determine how much carbon 14 (the radioactive isotope of carbon) is present in the artifact and comparing it to levels currently present in the atmosphere.Above is a graph that illustrates the relationship between how much Carbon 14 is left in a sample and how old it is.This discovery meant that there are three naturally occurring isotopes of carbon: Whereas carbon-12 and carbon-13 are stable isotopes, carbon-14 is unstable or radioactive.  $$So either the answer is that ridiculously big number (9.17e7) or 30,476 years, being calculated with the equation I provided and the first equation in your answer, respectively. A fossil found in an archaeological dig was found to contain 20% of the original amount of 14C. I do not get the -0.693 value, but perhaps my answer will help anyway. If we assume Carbon-14 decays continuously, then$$ C(t) = C_0e^, $$where C_0 is the initial size of the sample. Since it takes 5,700 years for a sample to decay to half its size, we know$$ \frac C_0 = C_0e^, $$which means$$ \frac = e^,  so the value of $C_0$ is irrelevant.

Working with several collaboraters, Libby established the natural occurrence of radiocarbon by detecting its radioactivity in methane from the Baltimore sewer.