A radioactive sample with a half-life of `1` month has the label : 'Activity=
`2` microcurie on 1-8-1991'. What would be its avtivity two months earlier?

Options

A. `1.0` microcurie
B. `0.5` microcurie
C. `4` microcurie
D. `8` microcurie

Answer

To solve the problem step by step, we will use the formula for radioactive decay, which relates the initial activity, final activity, and the number of half-lives that have passed.

### Step-by-Step Solution:

1. **Identify Given Values:**
- Half-life (T) = 1 month
- Activity on 1-8-1991 (A) = 2 microcurie
- Time difference (t) = 2 months (we want to find the activity 2 months earlier)

2. **Determine the Number of Half-Lives (n):**
- The number of half-lives (n) can be calculated using the formula:
\[ n = \frac{t}{T} \]
- Substituting the values:
\[ n = \frac{2 \text{ months}}{1 \text{ month}} = 2 \]

3. **Use the Activity Decay Formula:**
- The formula relating initial activity (A₀) and final activity (A) is:
\[ A = A_0 \left(\frac{1}{2}\right)^n \]
- Rearranging the formula to find the initial activity (A₀):
\[ A_0 = A \left(2^n\right) \]

4. **Substitute Known Values:**
- Now substitute A = 2 microcurie and n = 2 into the equation:
\[ A_0 = 2 \text{ microcurie} \times 2^2 \]
- Calculate \(2^2\):
\[ 2^2 = 4 \]
- Now substitute this back:
\[ A_0 = 2 \text{ microcurie} \times 4 = 8 \text{ microcurie} \]

5. **Conclusion:**
- The activity of the radioactive sample two months earlier would be **8 microcurie**.

### Final Answer:
The activity two months earlier is **8 microcurie**.

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