Sound intensity

Question:
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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