Question

The researchers found that the Neanderthal fossil had approximately 7.80 × 10–3 as much 14C as was present in the atmosphere around the living Neanderthal and the plants it ate.

Approximately how many half-lives have passed since the Neanderthal died?

1/128 half-lives
6 half-lives
7 half-lives
8 half-lives

Answer 1

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**.