Question

3. In order to understand the prehistory of the Hawaiian island of Lana'i better, anthropologists Maria Sweeney, Melinda Allen, and Boyd Dixon used radiocarbon dating on charcoal found in an ancient dwelling site, the Kaunolu Village National Historic Landmark, the largest archeological complex on the island. In one of their samples, they found that approximately 94% of the original carbon 14 remained. Using the fact that Carbon 14 decays by 1.202% every 100 years, determine the approximate age of this sample. In your word document, paste the Excel table you made to come up with your answer.

Answer 1

3. The radioactive decay formula is A = A₀ e^-kt where A₀ is the original quantity, A is the quantity remaining after t years and k is the constant of decay.

Here, Carbon 14 decays by 1.202% every 100 years so that if t = 100, then A = (1-0.01202)A₀ = 0.98798A₀. Hence, 0.98798A₀ = A₀ e^-100k or, e^-100k = 0.98798. Now, on taking natural log of both the sides, we get -100k = ln 0.98798 or, k = -(ln 0.98798)/100 = -0.012092824/100 = -0.00012092824.

Let the approximate age of the sample be t years. Then 0.94A₀ = A₀ e^{-0.00012092824 t} or, e^{-0.00012092824 t} = 0.94. Now, on taking natural log of both the sides, we get - 0.00012092824t = ln 0.94 = -0.061875403 so that t = 0.061875403 / 0.00012092824 = 511.67 ( on rounding off to 2 decimal places). Thus, the age of the sample is about 512 years ( on rounding off to the nearest hole number).