Question

problem 6:
during 4th of july, fireworks explode in the air, and the sound is propagated through air. if the total power of these waves is 250 kw, find the intensity of the wave 170 meters from the source. if the threshold of pain is 1w/m2, would this sound be painful at 170 meters from the source?

Explanation:

Step1: <Formula for intensity>

The formula for intensity \(I\) of a spherical wave is \(I=\frac{P}{4\pi r^{2}}\), where \(P\) is the power and \(r\) is the distance from the source.
Given \(P = 250\times10^{3}\space W\) and \(r = 170\space m\).

Step2: <Substitute values into formula>

Substitute \(P\) and \(r\) into the formula:
\(I=\frac{250\times 10^{3}}{4\pi\times(170)^{2}}\)
First calculate \((170)^{2}=28900\), then \(4\pi\times28900\approx4\times3.14\times28900 = 363032\)
\(I=\frac{250000}{363032}\approx0.69\space W/m^{2}\)

Step3: <Compare with pain threshold>

Compare \(I = 0.69\space W/m^{2}\) with the pain threshold \(I_{threshold}=1\space W/m^{2}\)

Answer:

The intensity of the wave is approximately \(0.69\space W/m^{2}\). Since \(0.69\space W/m^{2}<1\space W/m^{2}\), the sound would not be painful at \(170\) meters from the source.