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This problem involves the Doppler effect, which is the change in frequency or wavelength of a wave in relation to an observer moving relative to the source of the wave. Here, we are given:

- The fire truck emits a frequency of \( f_s = 450 \, \text{Hz} \),
- The difference in frequency detected by the observer (detector) is \( \Delta f = 58 \, \text{Hz} \),
- The speed of sound in air is \( v = 345 \, \text{m/s} \).

The observed frequency \( f_o \) when the fire truck is moving can be calculated using the Doppler shift formula:

\[
f_o = f_s \left( \frac{v}{v - v_s} \right)
\]

Where:
- \( v_s \) is the speed of the source (fire truck),
- \( f_s \) is the frequency emitted by the source,
- \( v \) is the speed of sound,
- \( f_o \) is the observed frequency.

Since the difference in frequency is 58 Hz, the observed frequency \( f_o \) can be written as:

\[
f_o = f_s + 58 = 450 + 58 = 508 \, \text{Hz}.
\]

Now, solving for \( v_s \) (the speed of the fire truck):

\[
508 = 450 \left( \frac{345}{345 - v_s} \right)
\]

We can now solve this equation for \( v_s \).The speed of the fire truck is approximately \( 39.39 \, \text{m/s} \).

Would you like more details, or do you have any questions? Here are five related questions to expand your understanding:

1. How does the Doppler effect work in light waves compared to sound waves?
2. What would happen to the observed frequency if the fire truck were moving away from the observer?
3. How does the speed of sound vary with temperature and how would that affect this problem?
4. What is the difference between redshift and blueshift in the context of the Doppler effect?
5. How is the Doppler effect applied in real-world technologies like radar or medical imaging?

**Tip**: The Doppler effect applies to any type of wave, including light, which is why it's used to study the motion of stars in astronomy!

A fire truck emitting a 450 Hz signal passes by a stationary detector. The difference in frequency measured by the detector is 58 Hz. If the speed of sound is 345 m/s, how fast is the fire truck moving?

Learn how to calculate the speed of a fire truck using the Doppler effect. The fire truck emits a 450 Hz signal, and a stationary detector measures a 58 Hz frequency shift. Using the Doppler formula and the speed of sound at 345 m/s, find the speed of the fire truck. Suitable for high school students in Grades 11-12.

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