A freshly prepared radioactive source of half-life `2 h` emits radiation of intensity which is 64 times the permissible safe level. The minimum time after which it would be possible to work safely with this source is

Options

A. 6h
B. 12h
C. 24h
D. 128h

Answer

To solve the problem, we need to determine the minimum time after which it would be safe to work with a radioactive source that has a half-life of 2 hours and is currently emitting radiation at an intensity 64 times the permissible safe level.

### Step-by-Step Solution:

1. **Understanding the Problem:**
- The radioactive source has a half-life of 2 hours.
- The current intensity of radiation is 64 times the permissible safe level.

2. **Determine the Safe Intensity Level:**
- To work safely, the intensity must be reduced to 1/64 of the original intensity.

3. **Using the Half-Life Formula:**
- The intensity after \( n \) half-lives can be expressed as:
\[ I = I_0 \left( \frac{1}{2} \right)^n \]
- Here, \( I_0 \) is the initial intensity, and \( I \) is the intensity after \( n \) half-lives.

4. **Setting Up the Equation:**
- We want to find \( n \) such that:
\[ I = \frac{I_0}{64} \]
- This leads to the equation:
\[ \frac{I_0}{64} = I_0 \left( \frac{1}{2} \right)^n \]

5. **Canceling \( I_0 \):**
- Since \( I_0 \) is common on both sides, we can simplify:
\[ \frac{1}{64} = \left( \frac{1}{2} \right)^n \]

6. **Expressing 64 as a Power of 2:**
- We know that:
\[ 64 = 2^6 \]
- Therefore, we can rewrite the equation as:
\[ \frac{1}{64} = \left( \frac{1}{2} \right)^6 \]

7. **Equating the Powers:**
- From the equation:
\[ \left( \frac{1}{2} \right)^n = \left( \frac{1}{2} \right)^6 \]
- We find that:
\[ n = 6 \]

8. **Calculating the Time:**
- Since each half-life is 2 hours, the total time \( T \) required for 6 half-lives is:
\[ T = n \times \text{half-life} = 6 \times 2 \text{ hours} = 12 \text{ hours} \]

### Final Answer:
The minimum time after which it would be possible to work safely with this source is **12 hours**.

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