Final answer:
The Doppler effect is used to explain the change in frequency of a sound wave when the source of the sound is moving relative to the observer. In this case, the frequency of the siren from the fire truck appears higher as it approaches and lower as it recedes. By using the formula for the Doppler effect and the given information, we can calculate the speed of the fire truck and the time it will take to reach the fire.
Step-by-step explanation:
The question describes the Doppler effect, which is the change in frequency or pitch of a sound wave when the source of the sound is moving relative to the observer. In this case, as the fire truck approaches you, the frequency of the siren that you hear is higher than the actual frequency of the siren, which is 450Hz. As the fire truck recedes from you, the frequency of the siren that you hear is lower than the actual frequency, which is 410Hz.
The Doppler effect can be calculated using the formula: Δf/f = v/c, where Δf is the change in frequency, f is the actual frequency, v is the velocity of the source relative to the observer, and c is the speed of sound. Rearranging this equation to solve for v, we have v = c * (Δf/f). Using the given frequencies of 450Hz and 410Hz, and the speed of sound of approximately 343m/s, we can calculate the change in frequency as Δf = f1 - f2 = 450Hz - 410Hz = 40Hz. Substituting these values into the equation, we have v = 343m/s * (40Hz/450Hz) = 30.36m/s.
Now, to find the time it takes for the fire truck to get from your position to the fire, we can use the formula: time = distance/velocity. The distance is given as 5.0km, which is equal to 5000m. Substituting the values, we have time = 5000m/30.36m/s = 164.51s. Therefore, it will take approximately 164.51 seconds for the fire truck to reach the fire, assuming it maintains a constant speed.