Final answer:
The frequency causing destructive interference, given the person stands 7.70 m from one speaker and 13.20 m from another, is approximately 31.18 Hz.
Step-by-step explanation:
To determine the frequency that causes destructive interference at the given points from two identical speakers, we use the concept of path difference and the speed of sound. Destructive interference occurs when the path difference between the sound waves from the two speakers is an odd multiple of half wavelengths.
In this case, for n = 1, the path length difference is one half of a wavelength, so the equation for path difference
ΔL = nλ/2 can be applied.
From the given distances, the path difference ΔL is
13.20 m - 7.70 m = 5.50 m.
Knowing that ΔL is also equal to λ/2 for n = 1, we can solve for the wavelength λ = 2ΔL = 11.00 m. The speed of sound in air is typically taken to be v = 343 m/s. We can then find the frequency using the relationship v = fλ, where f is the frequency. Substituting the known values, we get
f = v/λ = 343 m/s / 11.00 m.
Therefore, the frequency causing the destructive interference at the given distances from the two speakers is approximately 31.18 Hz.