For the stationary wave `y=4sin((pix)/(15))cos(96pit)`, the distance between a node and the next antinode is

Options

A. 7.5
B. 15
C. 22.5
D. 30

Answer

To find the distance between a node and the next antinode for the stationary wave given by the equation \( y = 4 \sin\left(\frac{\pi x}{15}\right) \cos(96 \pi t) \), we can follow these steps:

### Step 1: Identify the general form of the standing wave equation
The general form of a standing wave can be expressed as:
\[ y = A \sin\left(\frac{2\pi x}{\lambda}\right) \cos(2\pi ft) \]
where \( A \) is the amplitude, \( \lambda \) is the wavelength, and \( f \) is the frequency.

### Step 2: Compare the given equation with the general form
From the given equation \( y = 4 \sin\left(\frac{\pi x}{15}\right) \cos(96 \pi t) \), we can identify:
- The term \( \sin\left(\frac{\pi x}{15}\right) \) corresponds to \( \sin\left(\frac{2\pi x}{\lambda}\right) \).
- The term \( \cos(96 \pi t) \) corresponds to \( \cos(2\pi ft) \).

### Step 3: Determine the wavelength \( \lambda \)
From the comparison:
\[ \frac{2\pi}{\lambda} = \frac{\pi}{15} \]
To find \( \lambda \), we can rearrange the equation:
\[ \lambda = \frac{2\pi}{\frac{\pi}{15}} = 2 \times 15 = 30 \]

### Step 4: Calculate the distance between a node and the next antinode
The distance between a node and the next antinode is given by:
\[ \text{Distance} = \frac{\lambda}{4} \]
Substituting the value of \( \lambda \):
\[ \text{Distance} = \frac{30}{4} = 7.5 \]

### Conclusion
Thus, the distance between a node and the next antinode is \( 7.5 \) meters.