Answer:
r = 2.63 m
Explanation:
To find the distance at which the sound level is 120dB, you first calculate the intensity of the sound. You use the following formula:
[tex]\beta=10log(\frac{I}{I_o})[/tex] (1)
β: sound level = 120dB
I: intensity of the sound
Io: threshold of hearing = 10⁻12W/m^2
You solve the equation (1) for I and replace the values of all parameters:
[tex]\beta=log(\frac{I}{I_o})^{10}\\\\10^\beta=10^{log(\frac{I}{I_o})^{10}}=(\frac{I}{I_o})^{10}\\\\I=10^{\beta/10}I_o[/tex]
[tex]I=10^{120/10}(10^{-12}W/m^2)=1\frac{W}{m^2}[/tex]
Next, you use the following formula for the power of the sound with intensity I, and you solve for r:
[tex]I=\frac{P}{4\pi r^2}\\\\r=\sqrt{\frac{P}{4\pi I}}[/tex]
r: distance at which the sound level is 120dB
P: power of the sound = 87W
I: intensity of the sound = 1W/m^2
You replace the values of I and P for calculating r:
[tex]r=\sqrt{\frac{87W}{4\pi (1W/m^2)}}=2.63m[/tex]
The distance is at 2.63m from the source of the soundr = 2.63m
