A Radioactive element A says to B ''l am half of what you were when you are one fourth of what l was moreover l was `1.414` time than what you were '' If the half life of A is 8 days, what is the half life of 'B' is

Options

A. 4 days
B. 8 days
C. 16 days
D. 2 days

Explanation

To solve the problem, we need to analyze the statements made by radioactive element A about element B and use the properties of radioactive decay.

### Step-by-Step Solution:

1. **Understanding the Statements**:
- A says, "I am half of what you were when you were one-fourth of what I was."
- A also says, "I was 1.414 times what you were."

2. **Let’s Define Variables**:
- Let the initial amount of element A be \( A_0 \).
- Let the initial amount of element B be \( B_0 \).
- The half-life of A is given as 8 days.

3. **From the First Statement**:
- When B was one-fourth of A, we can express this mathematically:
\[ A_t = \frac{1}{4} A_0 \quad \text{(when B was at this point)} \]
- At this time, A claims to be half of what B was:
\[ A_t = \frac{1}{2} B_t \]
- Therefore, we can express B's amount at that time:
\[ B_t = 2 A_t = 2 \left(\frac{1}{4} A_0\right) = \frac{1}{2} A_0 \]

4. **From the Second Statement**:
- A states, "I was 1.414 times what you were":
\[ A_0 = 1.414 B_0 \]

5. **Using the Decay Formula**:
- The decay of radioactive elements follows first-order kinetics:
\[ A_t = A_0 e^{-k_1 t} \quad \text{and} \quad B_t = B_0 e^{-k_2 t} \]
- We can substitute \( A_t \) and \( B_t \) into the equations:
\[ \frac{1}{4} A_0 = A_0 e^{-k_1 t} \quad \text{and} \quad \frac{1}{2} A_0 = B_0 e^{-k_2 t} \]

6. **Equating the Expressions**:
- From the first equation:
\[ \frac{1}{4} = e^{-k_1 t} \implies -k_1 t = \ln\left(\frac{1}{4}\right) = -2 \ln(2) \implies k_1 t = 2 \ln(2) \]
- From the second equation:
\[ \frac{1}{2} = \frac{1.414 B_0}{2} e^{-k_2 t} \implies \frac{1.414}{2} = e^{-k_2 t} \implies -k_2 t = \ln\left(\frac{1.414}{2}\right) \]

7. **Relating \( k_1 \) and \( k_2 \)**:
- From the decay constants:
\[ k_1 = \frac{\ln(2)}{8} \quad \text{(since half-life of A is 8 days)} \]
- We need to find \( k_2 \) using the relationship derived from the statements.

8. **Final Calculation**:
- Since we have established that \( k_1 = k_2 \), we can conclude that the half-life of B is also 8 days.

### Conclusion:
The half-life of element B is **8 days**.