When two sound waves travel in the same direction in a medium, the displacement of a particle located at x at time t is given by
`y_1=0.05 cos (0.50 pi x-10 pi t) y_2=0.05 cos (0.46 pi x-92 pi t)`
where `y_1,y_2` and x in meters and t in seconds. The speed of sound in the medium is:

Options

A. `92m//s`
B. `200m//s`
C. `100m//s`
D. `332m//s`

Explanation

To find the speed of sound in the medium given the two sound wave equations, we can follow these steps:

### Step 1: Identify the wave equations
The two sound wave equations are given as:
1. \( y_1 = 0.05 \cos(0.50 \pi x - 10 \pi t) \)
2. \( y_2 = 0.05 \cos(0.46 \pi x - 9.2 \pi t) \)

### Step 2: Identify the wave parameters
From the standard wave equation \( y = A \cos(kx - \omega t) \), we can identify:
- For \( y_1 \):
- Amplitude \( A_1 = 0.05 \) m
- Wave number \( k_1 = 0.50 \pi \) rad/m
- Angular frequency \( \omega_1 = 10 \pi \) rad/s

- For \( y_2 \):
- Amplitude \( A_2 = 0.05 \) m
- Wave number \( k_2 = 0.46 \pi \) rad/m
- Angular frequency \( \omega_2 = 9.2 \pi \) rad/s

### Step 3: Calculate the speed of sound
The speed of sound \( v \) can be calculated using the relationship:
\[ v = \frac{\omega}{k} \]
We can use either wave equation to find the speed. Let's use the first wave equation:

- For \( y_1 \):
\[ v_1 = \frac{\omega_1}{k_1} = \frac{10 \pi}{0.50 \pi} \]

### Step 4: Simplify the equation
\[ v_1 = \frac{10 \pi}{0.50 \pi} = \frac{10}{0.50} = 20 \text{ m/s} \]

### Step 5: Calculate using the second wave equation
Now, let's calculate using the second wave equation:

- For \( y_2 \):
\[ v_2 = \frac{\omega_2}{k_2} = \frac{9.2 \pi}{0.46 \pi} \]

### Step 6: Simplify the equation
\[ v_2 = \frac{9.2 \pi}{0.46 \pi} = \frac{9.2}{0.46} \approx 20 \text{ m/s} \]

### Conclusion
Both calculations yield the same speed of sound in the medium. Therefore, the speed of sound in the medium is:
\[ \boxed{20 \text{ m/s}} \]