To solve this problem, let's break it down step by step:

Key Information:

1. Speed of sound: $v_{\text{sound}} = 340 \, \text{m/s}$
2. Time for car to reach the cyclist after honking: $t_{\text{car}} = 30 \, \text{s}$
3. Distance between car and cyclist when honk is heard: $d_{\text{sound}} = 1200 \, \text{m}$

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Step 1: Time for the sound to reach the cyclist

The time it takes for the sound of the honk to travel from the car to the cyclist is given by: $t_{\text{sound}} = \frac{d_{\text{sound}}}{v_{\text{sound}}}$

Substitute the values: $t_{\text{sound}} = \frac{1200}{340} \approx 3.53 \, \text{seconds.}$

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Step 2: Time for the car to reach the cyclist after sound is heard

The car takes $t_{\text{car}} = 30 \, \text{seconds}$ to reach the cyclist, but the cyclist hears the honk $t_{\text{sound}} \approx 3.53 \, \text{seconds}$ after it is made. Therefore, the total time from when the car honks to when it reaches the cyclist is: $T_{\text{car}} = t_{\text{car}} + t_{\text{sound}} \approx 30 + 3.53 = 33.53 \, \text{seconds.}$

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Step 3: Distance covered by the car

The car travels $d_{\text{car}} = 1200 \, \text{meters}$ in $T_{\text{car}} \approx 33.53 \, \text{seconds}.$

The speed of the car, $v_{\text{car}}$, is calculated as: $v_{\text{car}} = \frac{d_{\text{car}}}{T_{\text{car}}}$

Substitute the values: $v_{\text{car}} = \frac{1200}{33.53} \approx 35.8 \, \text{m/s.}$

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Final Answer:

The speed of the car is approximately: $\boxed{35.8 \, \text{m/s.}}$

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Would you like a detailed explanation of any step? Here are some related questions to expand your understanding:

1. How would the answer change if the speed of sound were different?
2. What if the car's speed was not constant? How could this problem be solved?
3. How do sound waves travel, and how does their speed vary with medium?
4. What if the cyclist started moving after hearing the honk? How would that affect the calculation?
5. Can this problem be solved using relative velocity concepts?

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Tip: In problems involving sound and motion, always account for the delay in sound reaching the observer when calculating total time