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`

Explanation

To solve the problem of finding 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 wave parameters
The equation of the stationary wave can be compared to the standard form:
\[ y = 2a \sin(kx) \cos(\omega t) \]
From the given equation, we can identify:
- Amplitude \( A = 4 \) (thus \( 2a = 4 \) implies \( a = 2 \))
- Wave number \( k = \frac{\pi}{15} \)
- Angular frequency \( \omega = 96\pi \)

### Step 2: Calculate the wavelength
The wave number \( k \) is related to the wavelength \( \lambda \) by the formula:
\[ k = \frac{2\pi}{\lambda} \]
Substituting the value of \( k \):
\[ \frac{\pi}{15} = \frac{2\pi}{\lambda} \]
Now, we can solve for \( \lambda \):
\[ \lambda = \frac{2\pi}{\frac{\pi}{15}} = 2 \times 15 = 30 \]

### Step 3: Determine the distance between a node and the next antinode
In a stationary wave, the distance between a node and the next antinode is given by:
\[ \text{Distance} = \frac{\lambda}{4} \]
Substituting the value of \( \lambda \):
\[ \text{Distance} = \frac{30}{4} = 7.5 \]

### Final Answer
The distance between a node and the next antinode is \( 7.5 \) units.

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