The half lives of a radioactive sample are 30 years and 60 years from `alpha-` emission and `beta-` emission respectively. If the sample decays both by `alpha-` emission and `beta-` emission emission simultaneously, the time after which only one-fourth of the sample remain is

Options

A. 10 years
B. 20 years
C. 40 years
D. 45 years

Answer

To solve the problem, we need to find the time after which only one-fourth of a radioactive sample remains, given that the sample decays through both alpha and beta emissions simultaneously.

### Step-by-Step Solution:

1. **Identify the Half-Lives**:
- The half-life for alpha emission, \( t_1 \), is 30 years.
- The half-life for beta emission, \( t_2 \), is 60 years.

2. **Calculate the Decay Constants**:
- The decay constant \( \lambda \) is related to the half-life by the formula:
\[ \lambda = \frac{\ln(2)}{t_{1/2}} \]
- For alpha emission:
\[ \lambda_1 = \frac{\ln(2)}{30 \text{ years}} \]
- For beta emission:
\[ \lambda_2 = \frac{\ln(2)}{60 \text{ years}} \]

3. **Calculate the Combined Decay Constant**:
- When both emissions occur simultaneously, the total decay constant \( \lambda_{total} \) is the sum of the individual decay constants:
\[ \lambda_{total} = \lambda_1 + \lambda_2 = \frac{\ln(2)}{30} + \frac{\ln(2)}{60} \]
- To add these fractions, find a common denominator (which is 60):
\[ \lambda_{total} = \frac{2\ln(2)}{60} + \frac{\ln(2)}{60} = \frac{3\ln(2)}{60} = \frac{\ln(2)}{20} \]

4. **Determine the Combined Half-Life**:
- The combined half-life \( t_{1/2, total} \) can be calculated from the total decay constant:
\[ t_{1/2, total} = \frac{\ln(2)}{\lambda_{total}} = \frac{\ln(2)}{\frac{\ln(2)}{20}} = 20 \text{ years} \]

5. **Calculate the Time for One-Fourth of the Sample to Remain**:
- To find the time required for the sample to reduce to one-fourth of its original amount, we need to consider that it takes two half-lives to achieve this:
\[ t = 2 \times t_{1/2, total} = 2 \times 20 \text{ years} = 40 \text{ years} \]

### Final Answer:
The time after which only one-fourth of the sample remains is **40 years**.

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