A normal adult jawbone contains 200 mg of Carbon-14 in a living person. If scientists found a jawbone that only had 50mg of Carbon-14, how old is the bone? (The half-life of C-14 is 5730 years).

Carbon-14 is a radioisotope of carbon that decays following first-order kinetics. There are four values of interest in this problem: the "normal" (or original) amount of carbon-14 for a jawbone (), the actual amount of carbon-14 in a jawbone (), the half-life of carbon-14 (), and the actual time elapsed () from the original time. There is an equation that ties all these values in together,

where k is the rate constant, which, for first-order decay, is related to the half-life by

What you want to find here is the time elapsed (t). So, you can substitute the latter equation for k into the k in the former equation to get

Rearranging to solve for t, the equation becomes

You are given all three of the values necessary to solve for t: The normal amount of carbon-14 is 200 mg; the actual amount of carbon-14 in the sample is 50 mg; and the half-life of carbon-14 is 5730 years. Plugging them into the above equation, we get

So the jawbone found is 11460 years old (or 11000 if accounting for sig figs).

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