For given values to `T` and `m`, resonance length in winter is `x_(1)` cm, and in summer, it is `x_(2)` cm. Then

Options

A. `x_(1) = x_(2)`
B. `x_(2) gt x_(1)`
C. `x_(2) lt x_(1)`
D. cannot say

Explanation

To solve the problem regarding the resonance lengths \( x_1 \) and \( x_2 \) in winter and summer respectively, we can follow these steps:

### Step-by-Step Solution:

1. **Understand Resonance Length**:
The resonance length in a resonance tube is given by the formula:
\[ L = (2n + 1) \frac{\lambda}{4} \]
where \( L \) is the resonance length, \( n \) is an integer (the mode number), and \( \lambda \) is the wavelength of the sound.

2. **Relate Wavelength to Velocity**:
The wavelength \( \lambda \) can be related to the velocity \( V \) of sound and frequency \( f \) by the equation:
\[ V = f \lambda \quad \Rightarrow \quad \lambda = \frac{V}{f} \]

3. **Velocity in Different Seasons**:
The speed of sound \( V \) in air is affected by temperature. Generally, the speed of sound increases with temperature. Therefore, we can say:
\[ V_{\text{summers}} > V_{\text{winters}} \]

4. **Relate Resonance Length to Velocity**:
Since the resonance length \( L \) is proportional to the wavelength \( \lambda \), and since \( \lambda \) is proportional to \( V \):
\[ L \propto \lambda \propto V \]
This means that if the velocity increases, the resonance length also increases.

5. **Conclusion**:
Given that the speed of sound is greater in summer than in winter, we can conclude:
\[ x_2 > x_1 \]
Therefore, the resonance length in summer \( x_2 \) is greater than the resonance length in winter \( x_1 \).

### Final Answer:
The resonance length in summer \( x_2 \) is greater than the resonance length in winter \( x_1 \).