Chapter 14: Problem 59

At a point \(15 \mathrm{m}\) from a source of spherical sound waves, you measure the intensity \(750 \mathrm{mW} / \mathrm{m}^{2}\). How far do you need to walk, directly away from the source, until the intensity is \(270 \mathrm{mW} / \mathrm{m}^{2} ?\)

Short Answer

Expert verified

To reduce the intensity of sound from \(750 \mathrm{mW} / \mathrm{m}^{2}\) to \(270 \mathrm{mW} / \mathrm{m}^{2}\), you would need to move to a distance of approximately \(21.66\) metres from the sound source.

Step by step solution

Identify The Equation Relating Sound Intensity And Distance

The intensity of sound is inversely proportional to the square of the distance from the sound source. This relationship is mathematically expressed as \(I = \frac{P}{4 \pi r^{2}}\), where \(I\) is the intensity of the sound, \(P\) is the power of the sound source, and \(r\) is the distance from the source.

Set Up The Ratio Of The Intensities

We are given both the initial and final intensities and the initial distance, and we're asked to find the final distance. Setting up an equation for the ratio of the intensities, we get: \[ \frac{I_{1}}{I_{2}} = \left( \frac{r_{2}}{r_{1}} \right)^{2}\] Substituting the given values, we get: \[ \frac{750}{270} = \left( \frac{r_{2}}{15} \right)^{2}\]

Solve For \(r_{2}\)

Solving for \(r_2\), we need to take the square root of both sides to get \(r_2\) on its own. \(r_{2}\) becomes: \[ r_{2}=15 \sqrt{\frac{750}{270}} \]

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At a point \(15 \mathrm{m}\) from a source of spherical sound waves, you measure the intensity \(750 \mathrm{mW} / \mathrm{m}^{2}\). How far do you need to walk, directly away from the source, until the intensity is \(270 \mathrm{mW} / \mathrm{m}^{2} ?\)

Short Answer

Step by step solution

Identify The Equation Relating Sound Intensity And Distance

Set Up The Ratio Of The Intensities

Solve For \(r_{2}\)

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