For a radioactive sample,`15/16` th part decays in 20 minutes. The half-life of the sample is

Options

A. 4 minutes
B. 5 minutes
C. 6 minutes
D. 7 minutes

Explanation

To solve the problem, we need to determine the half-life of a radioactive sample given that \( \frac{15}{16} \) of it decays in 20 minutes.

### Step-by-Step Solution:

1. **Identify the Initial Quantity**:
Let the initial quantity of the radioactive sample be \( N_0 \).

2. **Calculate Remaining Quantity**:
If \( \frac{15}{16} \) of the sample decays, then the remaining quantity after 20 minutes is:
\[ N = N_0 - \frac{15}{16} N_0 = \frac{1}{16} N_0 \]

3. **Relate Remaining Quantity to Half-Lives**:
We need to express \( \frac{1}{16} N_0 \) in terms of \( N_0 \) after a certain number of half-lives. We know that after each half-life, the quantity of the sample is halved.

We can express \( \frac{1}{16} \) as a power of \( \frac{1}{2} \):
\[ \frac{1}{16} = \left( \frac{1}{2} \right)^4 \]
This means that it takes 4 half-lives to reduce the sample from \( N_0 \) to \( \frac{1}{16} N_0 \).

4. **Determine Total Time for 4 Half-Lives**:
We know that the total time taken for this decay is 20 minutes. Since it takes 4 half-lives to reach \( \frac{1}{16} N_0 \), we can write:
\[ 4T_{1/2} = 20 \text{ minutes} \]
where \( T_{1/2} \) is the half-life.

5. **Calculate the Half-Life**:
To find \( T_{1/2} \), we can rearrange the equation:
\[ T_{1/2} = \frac{20 \text{ minutes}}{4} = 5 \text{ minutes} \]

Thus, the half-life of the sample is **5 minutes**.

### Final Answer:
The half-life of the sample is \( T_{1/2} = 5 \text{ minutes} \).

---