What is the expected change in decibels for Listener One who is 372 inches away compared to Listener Two at 204 inches?

To determine the expected change in decibels for Listener One at 372 inches compared to Listener Two at 204 inches, we need to consider the inverse square law of sound propagation. This law states that sound intensity decreases with the square of the distance from the source.

Firstly, we calculate the ratio of the distances:

[ \text{Distance Ratio} = \frac{372}{204} ]

Next, we convert the distance ratio into a change in sound intensity. The formula to convert this distance ratio to decibels is given by:

[ \Delta L = 20 \cdot \log_{10}\left(\frac{D_1}{D_2}\right) ]

Where (D_1) is the distance of Listener One and (D_2) is the distance of Listener Two.

Substituting the values, we compute the logarithm of the distance ratio:

1. Calculate the distance ratio:

[ \text{Distance Ratio} \approx 1.8235 ]

2. Now apply the formula:

[ \Delta L = 20 \cdot \log_{10}(1.8235) ]

This results in:

[ \Delta L \approx 20