Question
Given that the half-life of a certain radioactive isotope is 32 hours, find the fraction of the sample that would remain after 16 hours?
Question 9Select one:
0.62
0.71
0.5
0.25
Question 9Select one:
0.62
0.71
0.5
0.25
Answers
GPT-4o mini
To find the fraction of the sample that remains after a certain amount of time, we can use the half-life formula. The amount of material remaining after a certain time \( t \) can be calculated using the following formula:
\[
\text{Remaining fraction} = \left( \frac{1}{2} \right)^{\frac{t}{T_{1/2}}}
\]
where:
- \( T_{1/2} \) is the half-life of the substance,
- \( t \) is the time that has elapsed.
Given that the half-life \( T_{1/2} \) is 32 hours and we want to find the fraction remaining after 16 hours, we can substitute the values into the formula:
\[
\text{Remaining fraction} = \left( \frac{1}{2} \right)^{\frac{16 \, \text{hours}}{32 \, \text{hours}}} = \left( \frac{1}{2} \right)^{0.5}
\]
Calculating \( \left( \frac{1}{2} \right)^{0.5} \):
\[
\left( \frac{1}{2} \right)^{0.5} = \frac{1}{\sqrt{2}} \approx 0.7071
\]
Therefore, the fraction of the sample that would remain after 16 hours is approximately 0.7071, which is closest to:
**0.71**
\[
\text{Remaining fraction} = \left( \frac{1}{2} \right)^{\frac{t}{T_{1/2}}}
\]
where:
- \( T_{1/2} \) is the half-life of the substance,
- \( t \) is the time that has elapsed.
Given that the half-life \( T_{1/2} \) is 32 hours and we want to find the fraction remaining after 16 hours, we can substitute the values into the formula:
\[
\text{Remaining fraction} = \left( \frac{1}{2} \right)^{\frac{16 \, \text{hours}}{32 \, \text{hours}}} = \left( \frac{1}{2} \right)^{0.5}
\]
Calculating \( \left( \frac{1}{2} \right)^{0.5} \):
\[
\left( \frac{1}{2} \right)^{0.5} = \frac{1}{\sqrt{2}} \approx 0.7071
\]
Therefore, the fraction of the sample that would remain after 16 hours is approximately 0.7071, which is closest to:
**0.71**