To determine the age of the object, we can use the concept of half-lives. The half-life is the time it takes for half of the radioactive atoms to decay.
If 12.5% of the radioactive atoms remain, we can express this as a fraction:
\[ 12.5% = \frac{12.5}{100} = \frac{1}{8} \]
The fraction \(\frac{1}{8}\) can be expressed in terms of half-lives:
- After 1 half-life, 50% remains (which is \(\frac{1}{2}\)).
- After 2 half-lives, 25% remains (which is \(\frac{1}{4}\)).
- After 3 half-lives, 12.5% remains (which is \(\frac{1}{8}\)).
Since it takes 3 half-lives to reach 12.5% remaining, we can calculate the age of the object:
\[ \text{Age} = 3 \times \text{half-life} = 3 \times 800 \text{ years} = 2400 \text{ years} \]
Since the answer options provided are different, it appears there may have been an error in my calculations or in interpreting the choices. Let's double-check with the available options:
Given that only 12.5% remains after some time, the correct number of half-lives needed was indeed 3. Therefore,
\[ \text{Age} = 3 \times 800 \text{ years} = 2400 \text{ years} \]
However, since the options provided in your question did not include 2400 years, I may have misunderstood what was being asked. The closest approximation or reasoning does not seem to directly match the given options (3200, 6400, 1600).
It's also important to note from the given options, if we were to check:
- 1 half-life: 800 years (50% remaining)
- 2 half-lives: 1600 years (25% remaining)
- 3 half-lives: 2400 years (12.5% remaining, which is absent).
Therefore, while the calculated age of the sample is indeed 2400 years based on the mathematics of half-lives, none of the answer choices correlate directly with the needed age based on this decay concept.
Let me revise: If choosing based on closest available option, 1600 years is the computed age for 25% remaining, not that, if considered otherwise, the need is 3.
Please ensure that the question's options align with their possible output for half-lives. Would you like to proceed with further details or specification?
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