If `T= 2pisqrt(1/8)`, then relative errors in T and l are in the ratio

Options

A. `1//2`
B. `2`
C. `1//2pi`
D. none of these

Explanation

To solve the problem, we need to find the ratio of the relative errors in \( T \) and \( L \) given that \( T = 2\pi \sqrt{\frac{L}{G}} \).

### Step-by-Step Solution:

1. **Write the expression for \( T \)**:
\[ T = 2\pi \sqrt{\frac{L}{G}} \]

2. **Take the logarithm of both sides**:
\[ \log T = \log(2\pi) + \frac{1}{2} \log L - \frac{1}{2} \log G \]

3. **Differentiate both sides**:
Using the property of logarithms and differentiation, we have:
\[ \frac{dT}{T} = 0 + \frac{1}{2} \frac{dL}{L} - \frac{1}{2} \frac{dG}{G} \]
Since \( G \) is a constant, \( \frac{dG}{G} = 0 \). Thus, we simplify to:
\[ \frac{dT}{T} = \frac{1}{2} \frac{dL}{L} \]

4. **Relate differentials to errors**:
We can approximate differentials with errors:
\[ \frac{\Delta T}{T} \approx \frac{1}{2} \frac{\Delta L}{L} \]

5. **Express the ratio of relative errors**:
Rearranging gives us:
\[ \frac{\Delta T/T}{\Delta L/L} = \frac{1}{2} \]

6. **Conclusion**:
The ratio of the relative errors in \( T \) and \( L \) is:
\[ \frac{\Delta T/T}{\Delta L/L} = \frac{1}{2} \]

### Final Answer:
The ratio of the relative errors in \( T \) and \( L \) is \( \frac{1}{2} \).