To solve this problem correctly, you need to use the inverse square law of sound propagation. The inverse square law states that the intensity of sound decreases with the square of the distance from the source.
To calculate the distance from the source where the intensity level is 40 dB, follow these steps:
Step 1: Convert the decibel values to intensity ratios.
The formula to convert decibels (dB) to intensity ratios (I2/I1) is:
I2/I1 = 10^(L2-L1)/10
Here, L1 = 100 dB (initial intensity level) and L2 = 40 dB (desired intensity level)
I2/I1 = 10^(40 - 100)/10 = 10^(-60/10) = 10^(-6)
Step 2: Calculate the distance from the source.
The inverse square law states that the intensity ratio (I2/I1) is inversely proportional to the square of the distance (d1^2/d2^2):
I2/I1 = d1^2/d2^2
d2^2 = d1^2 * (I1/I2)
d2 = sqrt(d1^2 * (I1/I2))
Since we know the initial distance (d1) is 12 m, we can substitute it into the equation:
d2 = sqrt(12^2 * (1/10^-6))
d2 = sqrt(12^2 * 10^6)
d2 = sqrt(144 * 1,000,000)
d2 = 12 * 1000
d2 = 12,000 meters
So, the correct answer is that the distance from the source where the intensity level is 40 dB is 12,000 meters.
It appears that you made an error when calculating the ratio of the decibel values. Double-check your calculations, and make sure you use the correct formula for converting decibels to intensity ratios.