The equation of a stationary a stationary wave is represented by
`y=4sin((pi)/(6))(cos20pit)`
when x and y are in cm and t is in second.
Wavelength of the component waves is

Options

A. 4 cm
B. 20 cm
C. 12 cm
D. 6 cm

Answer

To find the wavelength of the component waves from the given stationary wave equation, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the given equation of the stationary wave**:
The equation is given as:
\[ y = 4 \sin\left(\frac{\pi}{6} x\right) \cos(20 \pi t) \]

2. **Compare with the standard form of a stationary wave**:
The standard form of a stationary wave is:
\[ y = 2a \sin(kx) \cos(\omega t) \]
Here, \( k \) is the wave number and \( \omega \) is the angular frequency.

3. **Extract the wave number \( k \)**:
From the equation, we can see that:
\[ k = \frac{\pi}{6} \]

4. **Use the relationship between wave number and wavelength**:
The wave number \( k \) is related to the wavelength \( \lambda \) by the formula:
\[ k = \frac{2\pi}{\lambda} \]

5. **Rearrange the formula to find \( \lambda \)**:
We can rearrange the formula to solve for \( \lambda \):
\[ \lambda = \frac{2\pi}{k} \]

6. **Substitute the value of \( k \)**:
Substitute \( k = \frac{\pi}{6} \) into the equation:
\[ \lambda = \frac{2\pi}{\frac{\pi}{6}} = 2\pi \cdot \frac{6}{\pi} \]

7. **Simplify the expression**:
The \( \pi \) cancels out:
\[ \lambda = 2 \cdot 6 = 12 \text{ cm} \]

8. **Conclusion**:
The wavelength of the component waves is:
\[ \lambda = 12 \text{ cm} \]

### Final Answer:
The wavelength of the component waves is \( 12 \) cm.

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