Final answer:
Using the Doppler effect formula and considering the bat's speed, the frequency of the sound that the bat hears reflected off the wall is approximately 40.21 kHz.
Step-by-step explanation:
When the bat is flying towards a wall, it would hear an increased frequency due to the Doppler effect, which is the change in frequency (or wavelength) of a wave in relation to an observer who is moving relative to the wave source. The formula to calculate this effect for a source moving towards a stationary observer (or, in this case, the reflective surface) is given by:
f' = \( \frac{f}{1 - \frac{v_s}{v}} \)
where:
- f' is the frequency heard by the bat,
- f is the emitted frequency (39.2 kHz),
- v_s is the speed of the source (bat) towards the wall (8.58 m/s), and
- v is the speed of sound in air, which is typically taken as 343 m/s at room temperature.
Substituting the values into the formula, we get:
f' = \( \frac{39.2 kHz}{1 - \frac{8.58 m/s}{343 m/s}} \)
Calculating this gives us the frequency that the bat would hear reflected from the wall:
f' = 39.2 kHz / (1 - 0.025) = 39.2 kHz / 0.975 ≈ 40.21 kHz.
Therefore, the frequency of the sound the bat hears reflected off the wall is approximately 40.21 kHz.