To find the distance of the speaker from each observer, we can use the inverse square law of sound.
The inverse square law states that the intensity of sound is inversely proportional to the square of the distance from the source of sound. Mathematically, it can be expressed as:
I1 / I2 = (r2 / r1)²
where I1 and I2 are the intensities of sound at distances r1 and r2 respectively.
First, let's convert the sound levels in decibels (dB) to intensities. The intensity of sound can be calculated using the following formula:
I = 10^(L/10)
where I is the intensity and L is the sound level in decibels.
For the observer recording a sound level of 51.9 dB, the intensity would be:
I1 = 10^(51.9/10) = 119.2263
For the observer recording a sound level of 82.5 dB, the intensity would be:
I2 = 10^(82.5/10) = 40237.3777
Now, let's substitute these intensities into the inverse square law equation:
119.2263 / 40237.3777 = (r2 / r1)²
To solve for the distances r1 and r2, we can take the square root of both sides of the equation:
√(119.2263 / 40237.3777) = r2 / r1
Now, let's calculate the square root and simplify the equation further:
0.019664656 ≈ r2 / r1
To find the actual distances, we can multiply this ratio by the known distance between the two observers (111 m):
0.019664656 * 111 = 2.183934216 ≈ r2
Therefore, the speaker is approximately 2.18 meters away from the observer with the sound level of 82.5 dB.
To find the distance of the speaker from the other observer, we can divide the known distance (111 m) by the ratio:
111 / 0.019664656 = 5637.519600 ≈ r1
Therefore, the speaker is approximately 5637.52 meters away from the observer with the sound level of 51.9 dB.