To determine the age of an object based on the remaining percentage of a radioactive element, we can use the half-life property of the element.
The half-life of the radioactive element is 800 years. If only 12.5% of the radioactive atoms remain, we can express this remaining amount as a fraction of the original amount:
12.5% = \( \frac{12.5}{100} = \frac{1}{8} \)
This means that the original amount of the element has gone through enough half-lives to reduce to \( \frac{1}{8} \) of its initial amount.
To find how many half-lives it takes to reduce to \( \frac{1}{8} \):
- \( \frac{1}{2} \) of the original amount \( \rightarrow \) After 1 half-life
- \( \frac{1}{4} \) of the original amount \( \rightarrow \) After 2 half-lives
- \( \frac{1}{8} \) of the original amount \( \rightarrow \) After 3 half-lives
Thus, it takes 3 half-lives to reach \( \frac{1}{8} \).
Now we calculate the total time:
Total time = Number of half-lives × Half-life duration
Total time = 3 × 800 years
Total time = 2400 years
Since the answer doesn't appear among the options you provided, let's go through the options:
- A. 6400 years
- B. 1600 years
- C. 3200 years
It looks like there was a miscalculation in the interpretation of the percentage reduction—let's re-evaluate:
If we reduce from 100% to 12.5%, we confirm we have achieved:
- 100% → 50% → 25% → 12.5%
This indeed confirms 3 half-lives.
Thus, the proper calculation stands with the solution being:
Total Age of the object = 2400 years
If we consider only given options, do check whether confirming from the course context is necessary. If forced to choose, the answer is not directly there. The logical reasoning leads to an age difference from what was initially considered.
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