An automobile moving at 30 m/s is approaching a factory whistle that has a frequency of 500 Hz. If the speed of the sound in air is 340 m/s, then the apparent frequency of the whistle as heard by the driver is

Options

A. 456 Hz
B. 544 Hz
C. 500 Hz
D. 597 Hz

Explanation

To find the apparent frequency of the whistle as heard by the driver of the automobile, we can use the Doppler effect formula for a stationary source and a moving observer. The formula is:

\[ f' = f \left( \frac{v + v_o}{v} \right) \]

Where:
- \( f' \) is the apparent frequency,
- \( f \) is the actual frequency of the source,
- \( v \) is the speed of sound in air,
- \( v_o \) is the speed of the observer (the automobile in this case).

### Step-by-Step Solution:

1. **Identify the given values:**
- Frequency of the whistle, \( f = 500 \, \text{Hz} \)
- Speed of sound in air, \( v = 340 \, \text{m/s} \)
- Speed of the observer (automobile), \( v_o = 30 \, \text{m/s} \)

2. **Substitute the values into the Doppler effect formula:**
\[ f' = 500 \left( \frac{340 + 30}{340} \right) \]

3. **Calculate the numerator:**
\[ 340 + 30 = 370 \, \text{m/s} \]

4. **Now substitute back into the formula:**
\[ f' = 500 \left( \frac{370}{340} \right) \]

5. **Calculate the fraction:**
\[ \frac{370}{340} \approx 1.0882 \]

6. **Now multiply by the original frequency:**
\[ f' \approx 500 \times 1.0882 \approx 544.1 \, \text{Hz} \]

7. **Round to the appropriate significant figures:**
\[ f' \approx 544 \, \text{Hz} \]

### Final Answer:
The apparent frequency of the whistle as heard by the driver is approximately **544 Hz**.