The car would need to approach at a minimum speed of 730.8 m/s (rounded to the nearest tenth) for the piccolo's sound to shift above your dog's hearing range while remaining audible to the driver
How to solve
Doppler shift formula: f' = f * (c + v) / (c - v), where:
f' is the perceived frequency
f is the original frequency (4186 Hz)
c is the speed of sound in air (343 m/s)
v is the relative velocity of the source (car)
Set f' at your dog's upper hearing limit (45,000 Hz):
45,000 Hz = 4186 Hz * (343 m/s + v) / (343 m/s - v)
Solve for v:
v = (45,000 Hz - 4186 Hz) * 343 m/s / (45,000 Hz + 4186 Hz) ≈ 730.8 m/s
Therefore, the car would need to approach at a minimum speed of 730.8 m/s (rounded to the nearest tenth) for the piccolo's sound to shift above your dog's hearing range while remaining audible to the driver
The Complete Question
You and your dog, with a maximum hearing range of 45,000 Hz, are standing on the side of the road. Someone approaches in a car playing an eighth-octave C on a piccolo (4186 Hz).
At what minimum speed (m/s) would the car have to approach for the piccolo's sound to shift above your dog's hearing range but remain audible to the car's driver (assume a normal human hearing range of 20-20,000 Hz)?
Calculate the speed rounded to the nearest tenth of a meter per second.
Parameters:
Speed of sound in air: 343 m/s
Piccolo frequency: 4186 Hz
Your hearing range: 20-20,000 Hz
Dog's hearing range: 20-45,000 Hz
Minimum frequency audible to driver: 20 Hz