Two sound sources produce progressive waves given by
`y_(1) = 6 cos 100 pi t`
and `y_(2) = 4 cos 102 pi t`
near the ears of an observer.
He will hear

Options

A. 4 beats per second and intensity ratio of maxima to minima ` = 3//2`
B. 2 beats per second and intensity ratio of maxima to minima ` = 9//4`
C. 1 beat per second and intensity ratio of maxima and minima ` = 25//1`
D. 1 beat per second and intensity ratio of maxima to minima ` = 9//4`

Explanation

To solve the problem, we need to analyze the given progressive waves produced by two sound sources and determine the number of beats per second and the intensity ratio.

### Step-by-Step Solution:

1. **Identify the given wave equations**:
We have two wave equations:
\[ y_1 = 6 \cos(100 \pi t) \]
\[ y_2 = 4 \cos(102 \pi t) \]

2. **Extract the angular frequencies**:
The general form of a wave is given by \( y = A \cos(\omega t) \), where \( \omega \) is the angular frequency. From the equations, we can identify:
- For \( y_1 \): \( \omega_1 = 100 \pi \)
- For \( y_2 \): \( \omega_2 = 102 \pi \)

3. **Calculate the frequencies**:
The frequency \( f \) is related to the angular frequency \( \omega \) by the formula \( f = \frac{\omega}{2\pi} \).
- For \( y_1 \):
\[ f_1 = \frac{100 \pi}{2 \pi} = 50 \text{ Hz} \]
- For \( y_2 \):
\[ f_2 = \frac{102 \pi}{2 \pi} = 51 \text{ Hz} \]

4. **Determine the number of beats per second**:
The number of beats per second is given by the absolute difference in frequencies:
\[ \text{Number of beats} = |f_1 - f_2| = |50 - 51| = 1 \text{ beat per second} \]

5. **Calculate the intensity ratio**:
The intensity \( I \) of a wave is proportional to the square of its amplitude \( A \). Therefore, the intensity ratio \( \frac{I_1}{I_2} \) is given by:
\[ \frac{I_1}{I_2} = \left(\frac{A_1}{A_2}\right)^2 \]
where \( A_1 = 6 \) and \( A_2 = 4 \):
\[ \frac{I_1}{I_2} = \left(\frac{6}{4}\right)^2 = \left(\frac{3}{2}\right)^2 = \frac{9}{4} \]

6. **Summarize the results**:
- Number of beats per second: \( 1 \)
- Intensity ratio: \( \frac{9}{4} \)

### Final Answer:
The observer will hear **1 beat per second** and the **intensity ratio** of maximum to minimum is **9:4**.