The half-life of a certain radioactive isotope is `32 h`.What fraction of a sample would ramain after `16h`?

Options

A. `0.25`
B. `0.71`
C. `0.29`
D. `0.75`

Explanation

To solve the problem of finding the fraction of a radioactive isotope remaining after 16 hours, given that its half-life is 32 hours, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Concept of Half-Life**:
The half-life of a radioactive isotope is the time required for half of the radioactive atoms in a sample to decay. In this case, the half-life is given as 32 hours.

2. **Determine the Number of Half-Lives**:
Since we want to find out how much of the sample remains after 16 hours, we first determine how many half-lives fit into this time frame:
\[ \text{Number of half-lives} = \frac{\text{Time elapsed}}{\text{Half-life}} = \frac{16 \text{ hours}}{32 \text{ hours}} = 0.5 \]

3. **Calculate the Remaining Fraction**:
The fraction of the sample remaining after a certain number of half-lives can be calculated using the formula:
\[ \text{Remaining fraction} = \left(\frac{1}{2}\right)^{\text{Number of half-lives}} \]
Substituting the number of half-lives we calculated:
\[ \text{Remaining fraction} = \left(\frac{1}{2}\right)^{0.5} = \frac{1}{\sqrt{2}} \]

4. **Numerical Value of the Remaining Fraction**:
The numerical value of \(\frac{1}{\sqrt{2}}\) can be calculated:
\[ \frac{1}{\sqrt{2}} \approx 0.707 \]

5. **Conclusion**:
Therefore, the fraction of the sample that remains after 16 hours is approximately \(0.707\).

### Final Answer:
The fraction of the sample that would remain after 16 hours is approximately **0.707**.