To determine approximately how many years ago the sample of wood was part of a living tree, we need to use the concept of the half-life of Carbon-14, which is the time it takes for half of the Carbon-14 to decay. For Carbon-14, the half-life is known to be 5,700 years.
Here is the step-by-step solution:
1. Initial Amount of Carbon-14: The sample initially contained 100 grams of Carbon-14.
2. Final Amount of Carbon-14: The sample now contains 25 grams of Carbon-14.
3. Determine the Number of Half-Lives:
- We need to find how many half-lives, denoted as [tex]\( n \)[/tex], have passed for the Carbon-14 amount to reduce from 100 grams to 25 grams.
- In one half-life, the amount of Carbon-14 would reduce to half of its initial amount. So, after the first half-life, the amount would be 50 grams; after the second half-life, it would be 25 grams.
- Therefore, it takes 2 half-lives for the amount to reduce from 100 grams to 25 grams.
4. Calculate the Time Elapsed:
- Since one half-life of Carbon-14 is 5,700 years, the time elapsed for 2 half-lives is:
[tex]\[
2 \text{ half-lives} \times 5,700 \text{ years/half-life} = 11,400 \text{ years}
\][/tex]
Thus, approximately 11,400 years ago, the sample was part of a living tree. The answer is:
(3) 11,400 yr
