Alright, let's solve this problem step-by-step. The half-life of argon-39 is 269 years, and we want to determine the fraction of the original amount of argon-39 remaining after 1,076 years.
1. Calculate the number of half-lives:
A half-life is the time required for half of a sample of a radioactive substance to decay. To find how many half-lives correspond to 1,076 years, we use the formula:
[tex]\[
\text{Number of half-lives} = \frac{\text{Time elapsed}}{\text{Half-life}}
\][/tex]
Substituting the given values:
[tex]\[
\text{Number of half-lives} = \frac{1076 \text{ years}}{269 \text{ years}} = 4
\][/tex]
So, 1,076 years is exactly 4 half-lives.
2. Calculate the remaining fraction:
After each half-life, half of the remaining argon-39 decays. Therefore, after [tex]\(n\)[/tex] half-lives, the fraction remaining is calculated by
[tex]\[
\left(\frac{1}{2}\right)^n
\][/tex]
Here, [tex]\( n = 4 \)[/tex]:
[tex]\[
\left(\frac{1}{2}\right)^4 = \frac{1}{16}
\][/tex]
3. Identify the correct answer:
The remaining fraction of the original amount of argon-39 after 1,076 years is [tex]\( \frac{1}{16} \)[/tex].
Therefore, the correct answer is:
[tex]\[
\boxed{\frac{1}{16}}
\][/tex]
The corresponding choice is:
C. [tex]$\frac{1}{16}$[/tex]
