if a person is 6.25 m away from a 60.0 w speaker, what is the sound level they are hearing? (treat the…

Explanation:

Step1: Calculate the intensity (I)

The formula for the intensity of a sound wave from a point source is (I=\frac{P}{4\pi r^{2}}), where (P = 60.0\space W) is the power of the source and (r=6.25\space m) is the distance from the source. [ \begin{align*} I&=\frac{60.0}{4\pi\times(6.25)^{2}}\ &=\frac{60.0}{4\pi\times39.0625}\ &=\frac{60.0}{156.25\pi}\ &\approx\frac{60.0}{490.87}\ &\approx 0.122\space W/m^{2} \end{align*} ]

Step2: Calculate the sound - level (\beta)

The formula for sound - level is (\beta=(10\space dB)\log_{10}\left(\frac{I}{I_{0}}\right)), where (I_{0} = 1\times10^{- 12}\space W/m^{2}) [ \begin{align*} \beta&=(10\space dB)\log_{10}\left(\frac{0.122}{1\times10^{-12}}\right)\ &=(10\space dB)\log_{10}(1.22\times10^{11})\ &=(10\space dB)(\log_{10}(1.22)+\log_{10}(10^{11}))\ &=(10\space dB)(0.086 + 11)\ &=(10\space dB)(11.086)\ &=110.86\space dB \end{align*} ]

Answer:

(111\space dB)