`""^(222)Rn` has a half-life period of 3.83 days. What fraction of the sample will remain undecoposed at the end of 10 days?

Answer

To solve the problem of how much of the sample of \(^{222}Rn\) will remain undecayed after 10 days, we can use the concept of half-life and the formula for radioactive decay.

### Step-by-Step Solution:

1. **Identify the Half-Life**:
The half-life (\(T_{1/2}\)) of \(^{222}Rn\) is given as 3.83 days.

2. **Determine the Total Time**:
We need to find out how much of the sample remains after 10 days.

3. **Calculate the Number of Half-Lives**:
To find the number of half-lives that fit into 10 days, we divide the total time by the half-life:
\[ \text{Number of half-lives} = \frac{10 \text{ days}}{3.83 \text{ days}} \approx 2.61 \]

4. **Use the Decay Formula**:
The fraction of the sample remaining after a certain number of half-lives can be calculated using the formula:
\[ \text{Fraction remaining} = \left(\frac{1}{2}\right)^{n} \]
where \(n\) is the number of half-lives. Plugging in our value:
\[ \text{Fraction remaining} = \left(\frac{1}{2}\right)^{2.61} \]

5. **Calculate the Fraction Remaining**:
Now we calculate:
\[ \text{Fraction remaining} \approx 0.164 \]

6. **Convert to Percentage**:
To express this fraction as a percentage, we multiply by 100:
\[ \text{Percentage remaining} = 0.164 \times 100 \approx 16.4\% \]

### Final Answer:
At the end of 10 days, approximately **16.4%** of the \(^{222}Rn\) sample will remain undecayed.