Final answer:
To find the minimum angle relative to the centerline perpendicular to the doorway at which someone outside the room will hear no sound, we can use the concept of diffraction. Using the formula θ = λ / a, where θ is the minimum angle, λ is the wavelength of the sound wave, and a is the width of the doorway, we can calculate the angle to be approximately 0.275 radians.
Step-by-step explanation:
To find the minimum angle relative to the centerline perpendicular to the doorway at which someone outside the room will hear no sound, we can use the concept of diffraction. Diffraction refers to the bending of waves as they pass through an opening or around an obstacle.
In this case, the sound waves with a frequency of 1200 Hz are passing through a doorway with a width of 1.03 mm. The minimum angle can be calculated using the formula:
θ = λ / a
where θ is the minimum angle, λ is the wavelength of the sound wave, and a is the width of the doorway.
First, we need to calculate the wavelength of the sound wave using the formula:
λ = v / f
where λ is the wavelength, v is the speed of sound (340 m/s), and f is the frequency of the sound wave (1200 Hz).
Substituting the values into the formula, we get:
λ = 340 m/s / 1200 Hz
λ ≈ 0.283 m
Next, we can substitute the values of λ and a into the formula for the minimum angle:
θ = 0.283 m / 1.03 mm
θ ≈ 0.275 radians
Therefore, someone outside the room will hear no sound when they are at a minimum angle of approximately 0.275 radians relative to the centerline perpendicular to the doorway.