To determine how much zinc remains in the fossil after 500 years given its half-life is 100 years, we can use the following formula:
\[ \text{Remaining amount} = \text{Initial amount} \times \left(\frac{1}{2}\right)^{\text{time elapsed} / \text{half-life}} \]
In this case:
- Initial amount = 200 g
- Time elapsed = 500 years
- Half-life = 100 years
First, calculate the number of half-lives that have passed in 500 years:
\[ \text{Number of half-lives} = \frac{500 \text{ years}}{100 \text{ years/half-life}} = 5 \]
Now we can calculate the remaining amount of zinc:
\[ \text{Remaining amount} = 200 , \text{g} \times \left(\frac{1}{2}\right)^5 \]
Calculating \(\left(\frac{1}{2}\right)^5\):
\[ \left(\frac{1}{2}\right)^5 = \frac{1}{32} \]
Now plug this value back into the formula:
\[ \text{Remaining amount} = 200 , \text{g} \times \frac{1}{32} = \frac{200}{32} \approx 6.25 , \text{g} \]
Thus, after 500 years, the amount of zinc remaining in the fossil is approximately 6.25 grams.
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