Question

The ratio of carbon-14 to nitrogen-14 in an artifact is 1:7. given that the half-life of carbon-14 is 5730 years, how old is the artifact?

a. 22,920 years
b. 17,190 years
c. 5730 years
d. 11,460 years

Can someone tell me the answer and how I can get to that that answer? I didn’t understand when my teacher explained it. What formula am I supposed to use?

Answer 1

Half-life formula: \(y=Ce^{kt}\)

You're given that the half-life of C-14 is 5730 years, which in terms of the formula gives you
\[\large\frac{1}{2}=e^{5730k}~~\Rightarrow~~k\approx -0.000121\]
C-14 decays into N-14. The fact that you're given a ratio of 1 to 7 means that only 1/8 of the initial C-14 remains. (Reason for this: 1 part C-14 and 7 parts N-14 mean there are 8 total parts.) As an equation, you have
\[\large \frac{1}{8}=e^{kt}~~\Rightarrow~~t\approx17190\]