Question

an archaeologist in Turkey discovers a spear head that contains 80% of its original amount of c-14. find the age of the spear to the nearest year

Answer 1

C-14 has a half-life of 5730 years, so we can model the amount of C-14 left in a sample using the exponential function: \(A(t)=A_0\left(\frac12\right)^{\frac{t}{5730}}\), which should make plenty of sense; if you plug in 5730 years, you get half of the original amount in the sample.

So, we're given that we have 80% of our original amount -- i.e. \(A(t)=0.8A_0\). Let's try to solve for \(t\):$$A(t)=A_0\left(\frac12\right)^{\frac{t}{5730}}\\
0.8A_0=A_0\left(\frac12\right)^{\frac{t}{5730}}\\
0.8=\left(\frac12\right)^{\frac{t}{5730}}$$ To get \(t\) alone, we're going to use a logarithm of base \(\frac12\):$$\log_{\frac12}0.8=\frac{t}{5730}\\
5730\log_{\frac12}0.8=t\\
t\approx1845$$ Thus our sample is approximately 1845 years old.

Answer 2

Thank you! ^_^