To determine how many hours have elapsed when only one-eighth of sodium-24 remains, we can use the concept of half-lives.
The half-life of sodium-24 is 15 hours. This means that every 15 hours, half of the remaining sodium-24 decays.
To find out how many half-lives it takes to go from a full amount to one-eighth, we can analyze the situation:
- After 1 half-life (15 hours), \( \frac{1}{2} \) of the original amount remains.
- After 2 half-lives (30 hours), \( \frac{1}{2} \times \frac{1}{2} = \frac{1}{4} \) remains.
- After 3 half-lives (45 hours), \( \frac{1}{4} \times \frac{1}{2} = \frac{1}{8} \) remains.
Since it takes 3 half-lives to reach one-eighth of the original amount, we can calculate the total elapsed time:
\[ 3 \text{ half-lives} \times 15 \text{ hours/half-life} = 45 \text{ hours} \]
Therefore, approximately 45 hours have elapsed.
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