Sure, let's solve this step-by-step.
1. Understanding Half-life:
The half-life of a substance is the time it takes for half of it to decay. For radium-226, this time is 1,600 years.
2. Determine the Number of Half-lives:
To find out how much radium-226 remains after 8,000 years, we need to determine how many half-lives have passed in that time period.
[tex]\[
\text{Number of half-lives} = \frac{\text{Total time period}}{\text{Half-life period}} = \frac{8,000 \text{ years}}{1,600 \text{ years}} = 5
\][/tex]
3. Calculating the Remaining Fraction:
Each half-life reduces the remaining amount of substance by half. After one half-life, [tex]\( \frac{1}{2} \)[/tex] of the original amount remains. After two half-lives, [tex]\( \left(\frac{1}{2}\right)^2 = \frac{1}{4} \)[/tex] remains, and this pattern continues. Therefore, after 5 half-lives, the remaining fraction can be calculated as follows:
[tex]\[
\text{Remaining fraction} = \left( \frac{1}{2} \right)^5 = \frac{1}{32}
\][/tex]
Thus, the fraction of the original amount of radium-226 that will still be radium after 8,000 years is [tex]\( \frac{1}{32} \)[/tex].
So, the correct answer is:
C. [tex]\( \frac{1}{32} \)[/tex]
