A train sounds its horn as it approaches an intersection. The horn can just be heard at a level of 53 dB by an observer 11 km away.
(a) What is the average power generated by the horn?
To find the average power generated by the horn, we need to use the equation that relates power to the sound intensity.
The sound intensity (I) at a certain distance from the source is given by the equation:
I = P / A
where I is the sound intensity, P is the power, and A is the area through which the sound waves pass.
In this case, the sound intensity is given as 53 dB, which is measured on the decibel scale. We need to convert it to the intensity in watts per square meter (W/m^2) using the equation:
I(W/m^2) = 10^(L/10)
where L is the sound intensity in decibels.
Converting 53 dB to intensity:
I(W/m^2) = 10^(53/10)
I(W/m^2) ≈ 0.001995 W/m^2
Now, we need to find the area through which the sound waves pass. In this case, the sound is heard by the observer 11 km away.
To calculate the area, we assume that the sound waves radiate uniformly in all directions, forming a sphere around the source. The surface area of a sphere is given by the equation:
A = 4πr^2
where A is the area and r is the distance from the source.
Converting the distance to meters:
11 km = 11,000 m
Calculating the area:
A = 4π(11,000)^2
A ≈ 1.52 x 10^9 m^2
Now we can substitute the values into the equation I = P / A to find the average power generated by the horn.
0.001995 = P / 1.52 x 10^9
Solving for P:
P ≈ 1.52 x 10^9 * 0.001995
P ≈ 3,028,440 W
Therefore, the average power generated by the horn is approximately 3,028,440 watts.