Final answer:
The frequency of the sound causing first-order constructive interference, when a person stands 3.75 m from one speaker and 7.25 m from another speaker, is 98 Hz.
Step-by-step explanation:
To find the frequency of sound that causes constructive interference, we need to consider the path difference traveled by sound from the two speakers to the point of interference. Constructive interference occurs when the path difference is an integral multiple of the wavelength. For the first order of constructive interference (n=1), the path difference is one wavelength (λ).
Let's denote d1 as the distance from the first speaker to the point of interference and d2 as the distance from the second speaker to the point of interference. The path difference Δd is therefore Δd = d2 - d1. In this case, the person is 3.75 m from one speaker and 7.25 m from another, so Δd = 7.25 m - 3.75 m = 3.5 m.
The wavelength (λ) is equal to the path difference for the first constructive interference (n=1), so λ = Δd = 3.5 m. To find the frequency (f), we use the equation v = fλ, where v is the speed of sound in air (assuming v = 343 m/s at room temperature). Hence, f = v/λ.
f = 343 m/s / 3.5 m = 98 Hz
Therefore, the frequency of the sound for the constructive interference at the given distances is 98 Hz.