There is a speaker emitting sound with a uniform power of 250 W. At what distance will the intensity be just below the threshold of pain, which is 1.00 W/m^2?
Answer:
The sound intensity is given by
I = \frac{P}{A}
The sound emitted uniformly in space, so it is spherically propagated. The area of sphere is 4\pi r^2. Thus,
I=\frac{P}{4\pi r^2}
Solve for the distance.
r^2 = \frac{P}{4\pi I} \\ r = \sqrt{\frac{P}{4\pi I}}
Plug in the numbers.
r = \sqrt{\frac{250 (\mathrm{W})}{4\pi \times 1.00 (\mathrm{W/m}^2)}}
Therefore, the distance is
r = 4.46 \mathrm{m}
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