To determine the fraction of potassium-40 remaining after 3.9 billion years, we can use the formula for radioactive decay, which relates the remaining quantity of a substance to its initial quantity and the number of half-lives that have elapsed.
- The half-life of potassium-40 is 1.3 billion years.
- The time that has passed is 3.9 billion years.
First, calculate the number of half-lives that have elapsed:
\[ \text{Number of half-lives} = \frac{\text{Time passed}}{\text{Half-life}} = \frac{3.9 \text{ billion years}}{1.3 \text{ billion years}} \approx 3 \]
Now, use the fact that after each half-life, the amount of substance remaining is halved. Therefore, after 3 half-lives, the fraction of the original quantity remaining is:
\[ \left(\frac{1}{2}\right)^3 = \frac{1}{8} \]
Thus, the fraction of original potassium-40 remaining after 3.9 billion years is \( \frac{1}{8} \).
The correct answer is:
D. \(\frac{1}{8}\)