Question 4.4.8.3: estimating the age of ancient Tools Traces of burned wood al......
estimating the age of ancient Tools
Traces of burned wood along with ancient stone tools an archeological dig in Chile were found to contain approximately 1.67% of the original amount of carbon 14. If the half-life of carbon 14 is 5730 years, approximately when was the tree cut and burned?
Using formula (3), the amount A of 14 present at time t is
A\left(t\right)\,=\,A_{0}e^{k t} k < 0 (3)
where A_{0} is the original amount of carbon 14 persent and k is a negative number. We first seek the number k. To find it, we use the fact that after 5700 years, half of the original amount of carbon 14 remains, so {A}\left(5730\right)\,=\frac12{A}_{0}. Then
{\frac{1}{2}}A_{0}=A_{0}e^{k(5730)}{\frac{1}{2}}=e^{5730k} Divide both sides of the equation by A_{0}
5730k=\ln{\frac{1}{2}} Rewrite as a logarithm.
k={\frac{1}{5730}}\ln{\frac{1}{2}}\approx -0.000120968Formula (3), therefore, becomes
A\left(t\right)\,=\,A_{0}e^{-0.000120968t}If the amount A of carbon 14 now present is 1.67% of the original amount, it follows that
0.0167A_{0}=A_{0}e^{-0.000120968t} \\ 0.0167 = e^{-0.000120968t} Divide both sides of the equation by A_{0}
0.000120968t = \ln0.0167 Rewrite as a logarithm
t={\frac{\ln0.0167}{-0.000120968}}\approx33,830\,\mathrm{years}The tree was cut and burned about 33,830 years ago. Some archeologists use this conclusion to argue that humans lived in the Americas nearly 34,000 years ago, much earlier than is generally accepted.