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Problem

A police car siren measures 70 dB to someone 44 meters away. What is the sound level of the police car is the police car is 1 meter away from the person? Round to the nearest decibel.

Police siren

Solution

There are two formulas that are useful for this problem.

1) $ \beta = 10 \log \left(\frac{I}{I_0}\right) $

where $ I_0 = {10}^{{10}^{-12}} \: \frac{W}{m^2} $

2) $ P = 4\pi r^2 I $

First, solve for the intensity at 44 meters.

$ 70 = 10 \log \left(\frac{I}{{}^{-12}}\right) $
$ I = (10^7)({10}^{-12}) = {10}^{-5} \: \frac{W}{m^2} $

Second, solve for the power, which is the same at any distance away from the sound source.

$ P = 4\pi ({10}^{-5}) {(44)}^{2} $
$ P = 7744 {(10)}^{-5} \pi \: W $

Third, solve for the intensity at 1 meter.

$ I = \frac{7744 {(10)}^{-5} \pi}{4\pi {(1)}^{2}} $
$ I = 1936 {(10)}^{-5} \: \frac{W}{m^2} $

Lastly, calculate the sound level in decibels.

$ \beta = 10\log \left(\frac{1936 {(10)}^{-5}}{{10}^{-12}}\right) $
$ \beta = 103 \: dB $
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