An archeological artifact was subjected to radiocarbon dating.The artifact showed a carbon-14 decay rate of 13.8 disintegrations/min per gram of carbon.Carbon-14 has a half-life of 5715 years,and currently living organisms decay at the rate of 15.3 disintegrations/min per gram of carbon.What is the approximate age of the artifact?

Question

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The age of the artifact can be calculated using the formula for radioactive decay, which relates the decay rate of the sample to the decay rate of a modern reference and the half-life of the isotope. The formula is $ t = \frac{\ln(\frac{N_0}{N})}{\lambda} $, where $ t $ is the time, $ N_0 $ is the initial quantity of the substance, $ N $ is the quantity that still remains and hasn’t yet decayed after time $ t $, and $ \lambda $ is the decay constant. The decay constant $ \lambda $ can be calculated from the half-life ($ T_{1/2} $) using the formula $ \lambda = \frac{\ln(2)}{T_{1/2}} $. However, for a simpler approach suitable for this question, we can use the ratio of the decay rates and the half-life to estimate the age without directly calculating the decay constant.Given that the current living organisms decay at a rate of 15.3 disintegrations/min per gram and the artifact decays at a rate of 13.8 disintegrations/min per gram, we can use the formula for exponential decay, considering that the decay rate decreases over time as the carbon-14 decays. The formula simplifies to $ t = T_{1/2} \cdot \frac{\ln(N_0/N)}{\ln(2)} $, where $ N_0/N $ is the ratio of the decay rates (15.3/13.8 in this case).Plugging in the numbers: $ t = 5715 \cdot \frac{\ln(15.3/13.8)}{\ln(2)} $, we find that the age of the artifact is closest to the option D, 851 years old. This calculation is based on the principle that the amount of carbon-14 decreases over time, and by comparing the decay rate of the artifact to that of a modern standard, we can estimate how many half-lives (or portions thereof) have passed since the artifact was last part of a living organism.

An Archeological Artifact Was Subjected to Radiocarbon Dating