To solve this problem, we can use the concept of half-lives. Assuming we are dealing with a substance that has a known half-life, we can determine how many half-lives it takes for the sample to decompose to 12.5% of its original amount.
Since 12.5% is \( \frac{1}{8} \) of the original amount, we need to find out how many times the substance has been halved to reach this point.
- After 1 half-life: \( 100% \rightarrow 50% \)
- After 2 half-lives: \( 50% \rightarrow 25% \)
- After 3 half-lives: \( 25% \rightarrow 12.5% \)
This shows that it takes 3 half-lives to reduce the original amount to 12.5%.
Next, we need to find out the duration of one half-life. If we assume, for instance, that the half-life of the substance is known (let's say it is 5,730 years as is common for certain radioactive substances), we can calculate the total time:
\[ \text{Total time} = \text{Number of half-lives} \times \text{Duration of one half-life} \] \[ \text{Total time} = 3 \times 5,730 \text{ years} = 17,190 \text{ years} \]
So the answer is:
D) 17,190 years.
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