A source of sound (1200 Hz) emits uniformly in all directions. An observer 2.99 m from the source measures a sound level of 41.1 dB. Calculate the average power output of the source.
To calculate the average power output of the source, we need to use the equation:
Power = (4πr²)(I)
Where:
- Power is the average power output of the source,
- r is the distance from the source to the observer, and
- I is the intensity of the sound.
First, let's calculate the intensity of the sound. The formula to calculate sound intensity is:
I = (P / A)
Where:
- I is the intensity of the sound,
- P is the power of the sound source, and
- A is the area of a sphere centered on the sound source.
Rearranging the equation, we can calculate the power in terms of intensity:
P = (I * A)
To calculate A, we need to use the formula:
A = 4πr²
Now, let's calculate the values step by step:
1. Convert the sound level from decibels (dB) to intensity (I) using the formula:
I = 10^(L/10)
Where:
- L is the sound level in decibels.
In this case, L = 41.1 dB:
I = 10^(41.1/10)
2. Calculate the area A of the sphere with radius r:
A = 4π(2.99)^2
3. Finally, calculate the power output of the source using the equation:
P = I * A
Substitute the calculated values of I and A into the equation to find P, the average power output of the source.