Let's complete the table step-by-step, discussing how much fermium-253 remains at each specified time interval.
### Given:
- Half-life of Fermium-253: 3.0 days
- Initial amount of Fermium-253: 216 micrograms
### Explanation:
The remaining amount of a substance after a certain time period can be calculated using the formula:
[tex]\[ \text{Remaining Amount} = \text{Initial Amount} \times \left( \frac{1}{2} \right)^{\frac{\text{Elapsed Time}}{\text{Half-life}}} \][/tex]
For 3.0 days:
- Time Elapsed: 3.0 days
- Elapsed Time / Half-life: [tex]\( \frac{3.0}{3.0} = 1 \)[/tex]
[tex]\[ \text{Remaining Amount after 3.0 days} = 216 \times \left( \frac{1}{2} \right)^{1} = 216 \times 0.5 = 108.0 \text{ micrograms} \][/tex]
For 6.0 days:
- Time Elapsed: 6.0 days
- Elapsed Time / Half-life: [tex]\( \frac{6.0}{3.0} = 2 \)[/tex]
[tex]\[ \text{Remaining Amount after 6.0 days} = 216 \times \left( \frac{1}{2} \right)^{2} = 216 \times 0.25 = 54.0 \text{ micrograms} \][/tex]
For 9.0 days:
- Time Elapsed: 9.0 days
- Elapsed Time / Half-life: [tex]\( \frac{9.0}{3.0} = 3 \)[/tex]
[tex]\[ \text{Remaining Amount after 9.0 days} = 216 \times \left( \frac{1}{2} \right)^{3} = 216 \times 0.125 = 27.0 \text{ micrograms} \][/tex]
### Table:
[tex]\[
\begin{tabular}{|c|c|}
\hline \begin{tabular}{c}
Time \\
Elapsed
\end{tabular} & \begin{tabular}{c}
Amount \\
Remaining
\end{tabular} \\
\hline 3.0 days & 108.0 \text{ micrograms} \\
\hline 6.0 days & 54.0 \text{ micrograms} \\
\hline 9.0 days & 27.0 \text{ micrograms} \\
\hline
\end{tabular}
\][/tex]
So, the amounts remaining at the specified times are 108.0 micrograms after 3.0 days, 54.0 micrograms after 6.0 days, and 27.0 micrograms after 9.0 days.
