A wave is represented by the equation
`y = A sin (10 pi x + 15 pi t + pi//3)`
Where x is in metre and t is in second.
a wave travelling in the positive x-direction with a velocity of 1.5 m/s
a wave travelling in the negative x-direction with a velocity 1.5 m/s
a wave travelling in the negative x-direction with a wavelength of 0.2 m
a wave travelling in the positive x-direction with a wavelength 0.2 m

Options

A. a wave travelling in the positive x-direction with a velocity of 1.5 m/s
B. a wave travelling in the negative x-direction with a velocity 1.5 m/s
C. a wave travelling in the negative x-direction with a wavelength of 0.2 m
D. a wave travelling in the positive x-direction with a wavelength 0.2 m

Explanation

To solve the problem, we need to analyze the given wave equation and extract the necessary information regarding its velocity, direction of propagation, and wavelength.

The wave is represented by the equation:
\[ y = A \sin(10\pi x + 15\pi t + \frac{\pi}{3}) \]

### Step 1: Identify the coefficients
From the wave equation, we can identify:
- The coefficient of \( x \) (wave number \( k \)) is \( 10\pi \).
- The coefficient of \( t \) (angular frequency \( \omega \)) is \( 15\pi \).

### Step 2: Calculate the wave velocity
The velocity \( v \) of the wave can be calculated using the formula:
\[ v = \frac{\omega}{k} \]
Substituting the values of \( \omega \) and \( k \):
\[ v = \frac{15\pi}{10\pi} = \frac{15}{10} = 1.5 \, \text{m/s} \]

### Step 3: Determine the direction of wave propagation
The direction of wave propagation can be determined by the signs of the coefficients of \( x \) and \( t \):
- Since both coefficients \( 10\pi \) (for \( x \)) and \( 15\pi \) (for \( t \)) are positive, the wave travels in the negative \( x \)-direction.

### Step 4: Calculate the wavelength
The wavelength \( \lambda \) can be calculated using the relationship between wave number \( k \) and wavelength:
\[ k = \frac{2\pi}{\lambda} \]
Substituting the value of \( k \):
\[ 10\pi = \frac{2\pi}{\lambda} \]
Solving for \( \lambda \):
\[ \lambda = \frac{2\pi}{10\pi} = \frac{2}{10} = 0.2 \, \text{m} \]

### Summary of Results
1. **Wave Velocity**: \( 1.5 \, \text{m/s} \)
2. **Direction of Propagation**: Negative \( x \)-direction
3. **Wavelength**: \( 0.2 \, \text{m} \)

### Conclusion
Based on the calculations:
- The wave is traveling in the negative \( x \)-direction with a velocity of \( 1.5 \, \text{m/s} \).
- The wave has a wavelength of \( 0.2 \, \text{m} \).

Thus, the correct options are:
- A wave traveling in the negative \( x \)-direction with a velocity of \( 1.5 \, \text{m/s} \) (Option B).
- A wave traveling in the negative \( x \)-direction with a wavelength of \( 0.2 \, \text{m} \) (Option C).