Answer:
When two sound waves with the same frequency and amplitude interfere, they can either add up constructively, resulting in a louder sound, or cancel each other out destructively, resulting in no sound at all. Destructive interference occurs when the waves are out of phase by half a wavelength, which means that the distance between the two sources is equal to an odd multiple of half the wavelength.
In this case, the distance between the two sirens is 4.00 m, which is equal to one wavelength (λ) plus half a wavelength (λ/2) of the sound waves they emit. Therefore, the wavelength of the sound waves is λ = 4.00 m / 1.5 = 2.67 m.
To find the distance from the metal plate where the fire fighter can stand and hear destructive interference, we need to calculate the distance from the plate to the fire fighter that is equal to an odd multiple of half the wavelength. Let's call this distance "x".
If the fire fighter is standing in front of one siren, the distance from the plate to the fire fighter is:
d1 = x
If the fire fighter moves towards the plate by a distance of half the wavelength, the distance from the plate to the fire fighter becomes:
d2 = x - λ/2
The difference between these two distances must be an odd multiple of half the wavelength for destructive interference to occur:
d2 - d1 = -λ/2 = -(2.67 m / 2) = -1.335 m
Therefore, the fire fighter can stand at a distance of x = 1.335 m away from the metal plate and hear destructive interference.