Step 1: Understanding the Concept:
This problem involves the Doppler effect for sound waves. The apparent frequency heard by a listener changes when there is relative motion between the source of the sound and the listener. When the source moves away from the listener, the observed frequency decreases.
Step 2: Key Formula or Approach:
The general formula for the Doppler effect is:
\[ f' = f \left( \frac{v \pm v_L}{v \mp v_S} \right) \]
where:
- $f'$ is the observed frequency.
- $f$ is the source frequency.
- $v$ is the speed of sound.
- $v_L$ is the speed of the listener.
- $v_S$ is the speed of the source.
The sign convention is: use the top sign for motion towards each other and the bottom sign for motion away from each other.
In this case:
- The listener is static, so $v_L = 0$.
- The source is moving away from the listener, so we use the bottom sign in the denominator (+).
Step 3: Detailed Explanation:
Given:
- Source frequency, $f = 800$ Hz.
- Speed of sound, $v = 340$ m/s.
- Speed of the source, $v_S = 30$ m/s.
- Speed of the listener, $v_L = 0$ m/s.
The formula for a source moving away from a stationary listener is:
\[ f' = f \left( \frac{v}{v + v_S} \right) \]
Substitute the given values:
\[ f' = 800 \left( \frac{340}{340 + 30} \right) \]
\[ f' = 800 \left( \frac{340}{370} \right) \]
\[ f' = 800 \times \frac{34}{37} \]
\[ f' = \frac{27200}{37} \approx 735.13 \text{ Hz} \]
Looking at the options, 733.3 Hz is the closest value. The slight difference might be due to rounding in the problem's intended answer or values. Let's re-check the calculation. $27200 / 37 = 735.135...$.
Option (C) 733.3 Hz is close. Let's assume there might be a typo in the options or the question values. However, based on standard calculation, it is the most plausible answer.
Let's calculate $800 \times (340/370) = 800 \times 0.9189... = 735.1...$ Hz.
The value 733.3 Hz would be obtained if $f' = 800 \times (330 / 360) = 800 \times (11/12) \approx 733.3$. This suggests the intended values might have been $v=330$ and $v_S=30$. Using the given values, 735.13 Hz is the correct answer, and 733.3 Hz is the closest option.
Step 4: Final Answer:
The calculated frequency is approximately 735.1 Hz. The closest option is 733.3 Hz. Therefore, option (C) is the correct answer.