To determine how many half-lives have passed for the Neanderthal fossil, we can use the concept of radioactive decay, specifically carbon-14 dating. You can calculate the number of half-lives that have passed using the ratio of the remaining carbon-14 in the fossil to the original amount.
The formula for finding the remaining ratio after a certain number of half-lives (\(n\)) is given by:
\[ \left( \frac{1}{2} \right)^n \]
where \(n\) is the number of half-lives.
Given that the fossil has approximately \(7.80 \times 10^{-3}\) (or \(0.0078\)) of the original carbon-14 amount, we want to find \(n\) such that:
\[ \left( \frac{1}{2} \right)^n = 0.0078 \]
To solve for \(n\), we can take the logarithm of both sides:
\[ n = \log_{0.5}(0.0078) \]
Using the change of base formula for logarithms, we can transform it into:
\[ n = \frac{\log(0.0078)}{\log(0.5)} \]
Now, calculating each logarithm:
- \(\log(0.0078) \approx -2.107\)
- \(\log(0.5) \approx -0.301\)
Now, substituting these values into the equation, we get:
\[ n \approx \frac{-2.107}{-0.301} \approx 6.99 \]
This suggests that approximately 7 half-lives have passed.
Looking at the provided options:
- \(1/128\) half-lives
- \(6\) half-lives
- \(7\) half-lives (this one matches our calculation)
- \(8\) half-lives
The correct answer is 7 half-lives.
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