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At a distance of 1.7 cm, the horn will begin to cause pain.
The source is an isotropic point source.
Let the distance at which the horn is barely audible be
[tex]r _2.[/tex]
[tex]r _2= 17 km[/tex]
Let the distance at which the horn will cause pain to be
[tex]r _2.[/tex]
[tex]A _{phere} = 4\pi \times r ^{2} [/tex]
The intensity ratio between barely audible and painful threshold is,
[tex]I = \frac{P }{A}[/tex]
[tex] \frac{I _2}{I _1 } = \frac{\frac{P}{4\pi \times r ^{2} _2 } }{ \frac{ P}{4\pi \times r ^{2} _1 }} [/tex]
[tex] = ( \frac{r _1 }{r _2 } ) ^{2} [/tex]
[tex]10 ^{ - 12} \: is \: ratio \: of \: the \: intensity [/tex]
between barely audible and painful threshold.
So,
[tex] \frac{I_2 }{I _1 } =( \frac{r _1 }{r _2 } ) ^{2} [/tex]
[tex] \frac{I_2 }{I _1 } = 10 ^{ - 12} [/tex]
[tex]r _2 = 17000 \: m[/tex]
The distance at which the horn will begin to cause pain is,
[tex]r _1 = r _2 \sqrt{10 ^{ - 12} } [/tex]
[tex]r _1 = 17000 \sqrt{10 ^{ - 12} } [/tex]
[tex]r _1 = 0.017 \: m[/tex]
[tex]r _1 = 1.7 \: cm[/tex]
Therefore, at a distance of 1.7 cm horn will begin to cause pain.
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