An artifact is discovered at a certain site. If it has $67 \%$ of the carbon- 14 it originally contained, what is the approximate age of the artifact to the nearest year? (carbon-14 decays at the rate of $0.0125 \%$ annually.) A. 2640 years B. 5360 years C. 1391 years D. 3204 years

Step 1 ：The decay of carbon-14 can be modeled by the exponential decay formula: \( N = N_0 * e^{-kt} \) where \(N\) is the final amount of the substance, \(N_0\) is the initial amount of the substance, \(k\) is the decay constant, and \(t\) is the time elapsed.

Step 2 ：In this case, we know that \(N/N_0 = 0.67\) (the artifact has 67% of the carbon-14 it originally contained), and \(k = 0.000125\) (carbon-14 decays at the rate of 0.0125% annually). We want to find \(t\), the age of the artifact.

Step 3 ：We can rearrange the formula to solve for \(t\): \( t = -\frac{1}{k} * \ln(N/N_0) \)

Step 4 ：Now we can plug in the given values and calculate \(t\).

Step 5 ：Final Answer: The approximate age of the artifact to the nearest year is \(\boxed{3204}\) years.