Question

An artifact was found and tested for its carbon-14 content. If 82% of the original carbon-14 was still present, What is its probable age (to the nearest 100 years)?

Answer 1

Uh, @killerbee, that isn't correct.

At t = 0, there's the full amount of carbon-14.
At t =5730 years, 1/2 of the full amount has decayed, leaving 1/2.
At t = 11460 years, 1/2 of the 1/2 has decayed, leaving 1/4.
At t = 3*5730 years, 1/2 of the 1/2 of the 1/2 has decayed, leaving 1/8
etc.

If \(C_0\) is the original amount of carbon-14, and the half-life is 5730 years, we can write \[C(t) = C_0(\frac{1}{2})^{(t/5730)}\] for the amount of carbon-14 remaining after \(t\) years.

Answer 2

We can evaluate this to find the answer without needing to know an absolute value of carbon-14 in the sample by dividing both sides by \(C_0\)
\[\frac{C(t)}{C_0} = (\frac{1}{2})^{(t/5730)}\]That will give us our fraction of the original, 0.82.
\[0.82 = (0.5)^{(t/5730)}\]Take the log of both sides, and remember that \(\log a^n = n\log a\). Solve for \(t\), and don't forget to round to the nearest 100 years.

Answer 3

Here's a graph of \(\dfrac{C(t)}{C_0}\) for 30,000 years: