In a resonance tube with tuning fork of frequency `512 Hz`, first resonance occurs at water level equal to `30.3cm` and second resonance ocuurs at `63.7cm`. The maximum possible error in the speed of sound is

Options

A. (a) `51.2cm//s`
B. (b) `102.4cm//s`
C. ( c ) 204.8cm//s`
D. (d) `153.6cm//s`

Explanation

To solve the problem of finding the maximum possible error in the speed of sound based on the given resonance tube data, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Resonance Levels:**
- First resonance occurs at \( L_1 = 30.3 \, \text{cm} \)
- Second resonance occurs at \( L_2 = 63.7 \, \text{cm} \)

2. **Calculate the Difference in Resonance Levels:**
\[ L_2 - L_1 = 63.7 \, \text{cm} - 30.3 \, \text{cm} = 33.4 \, \text{cm} \]

3. **Relate the Resonance Levels to Wavelength:**
- The first resonance corresponds to a quarter wavelength (\( \lambda/4 \)).
- The second resonance corresponds to three-quarters of the wavelength (\( 3\lambda/4 \)).
- The difference between the two resonance levels can be expressed as:
\[ L_2 - L_1 = \frac{\lambda}{2} \]
Hence,
\[ \lambda = 2(L_2 - L_1) = 2 \times 33.4 \, \text{cm} = 66.8 \, \text{cm} = 0.668 \, \text{m} \]

4. **Calculate the Speed of Sound:**
- The speed of sound \( V \) can be calculated using the formula:
\[ V = f \lambda \]
where \( f = 512 \, \text{Hz} \).
\[ V = 512 \, \text{Hz} \times 0.668 \, \text{m} = 342.656 \, \text{m/s} \]

5. **Determine the Error in Length Measurements:**
- The least count error in measuring the water level is given as \( \Delta L = 0.1 \, \text{cm} = 0.001 \, \text{m} \).
- Since both \( L_1 \) and \( L_2 \) have the same error, we can use the formula for the error in speed \( \Delta V \):
\[ \Delta V = 2f \Delta L \]
Substituting the values:
\[ \Delta V = 2 \times 512 \, \text{Hz} \times 0.001 \, \text{m} = 1.024 \, \text{m/s} \]

6. **Final Calculation of Maximum Possible Error in Speed of Sound:**
- The maximum possible error in the speed of sound is:
\[ \Delta V \approx 1.024 \, \text{m/s} \approx 1.024 \times 100 \, \text{cm/s} = 102.4 \, \text{cm/s} \]

### Conclusion:
The maximum possible error in the speed of sound is approximately \( 102.4 \, \text{cm/s} \).