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$$So either the answer is that ridiculously big number (9.17e7) or 30,476 years, being calculated with the equation I provided and the first equation in your answer, respectively. A fossil found in an archaeological dig was found to contain 20% of the original amount of 14C. I do not get the -0.693 value, but perhaps my answer will help anyway. If we assume Carbon-14 decays continuously, then$$ 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.