Answer
We can use the inverse square law for sound to determine the relationship between distance and sound intensity. The inverse square law states that sound intensity is inversely proportional to the square of the distance from the source:
I_1 / I_2 = (r_2 / r_1)^2
where I_1 and I_2 are the intensities at distances r_1 and r_2, respectively.
We can rearrange this equation to solve for r_2:
r_2 = r_1 * sqrt(I_1 / I_2)
where sqrt denotes the square root function.
We are given that the intensity level at a distance of 12.0 m is 70 dB. We can use the formula for converting between intensity and intensity level to find the intensity at this distance:
IL = 10 log(I / I_ref)
70 dB = 10 log(I / (1 x 10^-12 W/m^2))
I = 1.0 x 10^-6 W/m^2
We want to find the distance at which the intensity level is 40 dB. We can use the same formula to convert the intensity level to intensity:
40 dB = 10 log(I / (1 x 10^-12 W/m^2))
I = 1.0 x 10^-10 W/m^2
Now we can substitute the values for I_1 and I_2 into the equation for r_2:
r_2 = 12.0 m * sqrt((1.0 x 10^-6 W/m^2) / (1.0 x 10^-10 W/m^2))
r_2 = 120.0 m
Therefore, the distance from the point source where the intensity level is 40 dB is 120.0 m.
Work Step by Step
We can use the inverse square law for sound to determine the relationship between distance and sound intensity. The inverse square law states that sound intensity is inversely proportional to the square of the distance from the source:
I_1 / I_2 = (r_2 / r_1)^2
where I_1 and I_2 are the intensities at distances r_1 and r_2, respectively.
We can rearrange this equation to solve for r_2:
r_2 = r_1 * sqrt(I_1 / I_2)
where sqrt denotes the square root function.
We are given that the intensity level at a distance of 12.0 m is 70 dB. We can use the formula for converting between intensity and intensity level to find the intensity at this distance:
IL = 10 log(I / I_ref)
70 dB = 10 log(I / (1 x 10^-12 W/m^2))
I = 1.0 x 10^-6 W/m^2
We want to find the distance at which the intensity level is 40 dB. We can use the same formula to convert the intensity level to intensity:
40 dB = 10 log(I / (1 x 10^-12 W/m^2))
I = 1.0 x 10^-10 W/m^2
Now we can substitute the values for I_1 and I_2 into the equation for r_2:
r_2 = 12.0 m * sqrt((1.0 x 10^-6 W/m^2) / (1.0 x 10^-10 W/m^2))
r_2 = 120.0 m
Therefore, the distance from the point source where the intensity level is 40 dB is 120.0 m.
