The half-life of `^215At` is `100mus`. The time taken for the activity of a sample of `^215At` to decay to `1/16th` of its initial value is

Options

A. (a) `400 mus`
B. (b) `63mus`
C. (c) `40 mus`
D. (d) `300 mus`

Explanation

To solve the problem, we need to determine the time taken for the activity of a sample of `^215At` to decay to `1/16th` of its initial value, given that the half-life of `^215At` is `100 microseconds`.

### Step-by-Step Solution:

1. **Understand the concept of half-life**:
The half-life of a radioactive substance is the time required for half of the substance to decay. In this case, the half-life of `^215At` is `100 microseconds`.

2. **Determine the number of half-lives for the activity to decay to `1/16th`**:
We know that:
\[ \text{Remaining Activity} = \frac{A_0}{2^n} \]
where \( A_0 \) is the initial activity and \( n \) is the number of half-lives.

We want the remaining activity to be \( \frac{A_0}{16} \). Therefore, we set up the equation:
\[ \frac{A_0}{16} = \frac{A_0}{2^n} \]
This simplifies to:
\[ 16 = 2^n \]

3. **Solve for \( n \)**:
We can express 16 as a power of 2:
\[ 16 = 2^4 \]
Thus, we have:
\[ n = 4 \]

4. **Calculate the total time taken**:
Since each half-life is `100 microseconds`, the total time \( T \) taken for the activity to decay to `1/16th` of its initial value is:
\[ T = n \times \text{half-life} = 4 \times 100 \text{ microseconds} = 400 \text{ microseconds} \]

### Final Answer:
The time taken for the activity of the sample of `^215At` to decay to `1/16th` of its initial value is **400 microseconds**.

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