Two cars are moving on two perpendicular roads towards a crossing with uniform speeds of `72(km)/(h)` and `36(km)/(h)`. If second car blows born of frequency 280 Hz, then the frequency of horn heard by the driver of first car when the line joining the cars makes angle of `45^@C` with the roads, will be (velocity of sound is `330(m)/(s)`)

Options

A. (a)321 Hz
B. (b)298 Hz
C. (c)289 Hz
D. (d)280 Hz

Explanation

To solve the problem step by step, we need to apply the principles of the Doppler Effect. Here’s how we can approach the solution:

### Step 1: Convert Speeds from km/h to m/s
The speeds of the cars are given in km/h. We need to convert them to m/s for consistency with the velocity of sound.

- Speed of the first car (V1) = 72 km/h = \( \frac{72 \times 1000}{3600} = 20 \, \text{m/s} \)
- Speed of the second car (V2) = 36 km/h = \( \frac{36 \times 1000}{3600} = 10 \, \text{m/s} \)

### Step 2: Determine the Components of Velocity
Since the cars are moving towards each other at a 45-degree angle, we need to find the components of their velocities along the line joining them.

- For the second car (source of sound):
- \( V_s = V2 \cos(45^\circ) = 10 \cdot \frac{1}{\sqrt{2}} = \frac{10}{\sqrt{2}} \approx 7.07 \, \text{m/s} \)

- For the first car (observer):
- \( V_o = V1 \cos(45^\circ) = 20 \cdot \frac{1}{\sqrt{2}} = \frac{20}{\sqrt{2}} \approx 14.14 \, \text{m/s} \)

### Step 3: Apply the Doppler Effect Formula
The formula for the observed frequency \( f' \) when the source and observer are moving towards each other is given by:

\[ f' = f \frac{v + V_o}{v - V_s} \]

Where:
- \( f \) = frequency of the source = 280 Hz
- \( v \) = speed of sound = 330 m/s
- \( V_o \) = component of the observer's velocity = \( 14.14 \, \text{m/s} \)
- \( V_s \) = component of the source's velocity = \( 7.07 \, \text{m/s} \)

Substituting the values into the formula:

\[ f' = 280 \frac{330 + 14.14}{330 - 7.07} \]

### Step 4: Calculate the Observed Frequency
Now, we calculate the numerator and denominator:

- Numerator: \( 330 + 14.14 = 344.14 \)
- Denominator: \( 330 - 7.07 = 322.93 \)

Now substituting these values back into the equation:

\[ f' = 280 \frac{344.14}{322.93} \]

Calculating this gives:

\[ f' \approx 280 \cdot 1.067 = 298.76 \, \text{Hz} \]

### Step 5: Round to the Nearest Whole Number
The final frequency heard by the driver of the first car is approximately:

\[ f' \approx 299 \, \text{Hz} \]

### Conclusion
Thus, the frequency of the horn heard by the driver of the first car is approximately **299 Hz**.