To solve the problem of finding the reading of the water level in the resonance column when the first resonance occurs, we can follow these steps:
### Step-by-Step Solution:
1. **Understanding Resonance in a Column**:
The first resonance in a closed column occurs when the length of the air column (L) is equal to one-fourth of the wavelength (λ) of the sound wave. This can be expressed as:
\[
L = \frac{\lambda}{4}
\]
2. **Incorporating the End Correction**:
The effective length of the column must account for the end correction (e), which is given by:
\[
e = 0.6 \times r
\]
where \( r \) is the radius of the tube. Given that the diameter of the column tube is 4 cm, the radius \( r \) is:
\[
r = \frac{4 \, \text{cm}}{2} = 2 \, \text{cm} = 0.02 \, \text{m}
\]
Thus, the end correction \( e \) becomes:
\[
e = 0.6 \times 0.02 \, \text{m} = 0.012 \, \text{m} = 1.2 \, \text{cm}
\]
3. **Calculating Wavelength**:
The wavelength can be expressed in terms of the effective length:
\[
\lambda = 4(L + e)
\]
Substituting the end correction into the equation gives:
\[
\lambda = 4(L + 1.2 \, \text{cm})
\]
4. **Using the Speed of Sound**:
The speed of sound \( v \) is related to frequency \( f \) and wavelength \( \lambda \) by the equation:
\[
v = f \cdot \lambda
\]
Given that the speed of sound is \( 336 \, \text{m/s} \) and the frequency of the tuning fork is \( 512 \, \text{Hz} \), we can rearrange this to find \( \lambda \):
\[
\lambda = \frac{v}{f} = \frac{336 \, \text{m/s}}{512 \, \text{Hz}} = 0.65625 \, \text{m} = 65.625 \, \text{cm}
\]
5. **Finding the Length of the Column**:
Now we can substitute \( \lambda \) back into the equation for \( \lambda \):
\[
65.625 \, \text{cm} = 4(L + 1.2 \, \text{cm})
\]
Solving for \( L \):
\[
L + 1.2 \, \text{cm} = \frac{65.625 \, \text{cm}}{4} = 16.40625 \, \text{cm}
\]
\[
L = 16.40625 \, \text{cm} - 1.2 \, \text{cm} = 15.20625 \, \text{cm}
\]
6. **Final Result**:
Rounding to one decimal place, the reading of the water level in the column when the first resonance occurs is:
\[
L \approx 15.2 \, \text{cm}
\]
### Conclusion:
The correct answer is (b) 15.2 cm.
