In an organ pipe whose one end is at x =0, the pressureis expressed by `P = P_(0) "cos" (3 pi x)/(2) sin 300 pi t` where x is in meter and t in sec. The organ pipe can be :-

Options

A. Closed at one end, open at another with length 0.5 m =
B. Open at both ends, length = 1 m
C. Closed at both ends, length 2 m
D. Closed at end, open at another with length `(2)/(3)m`

Answer

To determine the type of organ pipe based on the given pressure equation, we will analyze the pressure at different positions along the pipe. The pressure is given by:

\[ P = P_0 \cos\left(\frac{3\pi x}{2}\right) \sin(300\pi t) \]

where \( x \) is in meters and \( t \) is in seconds.

### Step-by-Step Solution:

1. **Identify the Closed End**:
- The closed end of the organ pipe will have maximum pressure, which occurs when the cosine term is equal to 1.
- Substitute \( x = 0 \) into the pressure equation:
\[ P = P_0 \cos\left(\frac{3\pi \cdot 0}{2}\right) \sin(300\pi t) = P_0 \cos(0) \sin(300\pi t) = P_0 \cdot 1 \cdot \sin(300\pi t) \]
- Since \( \cos(0) = 1 \), the pressure is maximum at \( x = 0 \).

2. **Check the Open End**:
- The open end of the organ pipe will have minimum pressure, which occurs when the cosine term is equal to 0.
- Substitute \( x = 1 \) into the pressure equation:
\[ P = P_0 \cos\left(\frac{3\pi \cdot 1}{2}\right) \sin(300\pi t) = P_0 \cos\left(\frac{3\pi}{2}\right) \sin(300\pi t) \]
- Since \( \cos\left(\frac{3\pi}{2}\right) = 0 \), the pressure is minimum at \( x = 1 \).

3. **Check for Maximum Pressure Again**:
- Now, check at \( x = 2 \):
\[ P = P_0 \cos\left(\frac{3\pi \cdot 2}{2}\right) \sin(300\pi t) = P_0 \cos(3\pi) \sin(300\pi t) \]
- Since \( \cos(3\pi) = -1 \), the pressure is maximum (but negative) at \( x = 2 \).

4. **Conclusion**:
- We have found that at \( x = 0 \) (closed end), the pressure is maximum, and at \( x = 1 \) (open end), the pressure is minimum. At \( x = 2 \), the pressure is also maximum (but negative).
- This indicates that the organ pipe is closed at both ends.

### Final Answer:
The organ pipe can be classified as a **closed organ pipe**.