Answer
(a) The intensity at a distance of 1000 meters is $2.5\times 10^{-4}~W/m^2$
(b) The maximum distance at which the siren can be heard is 16 km
Work Step by Step
(a) We can find the intensity $I_2$ at 1000 m.
$\frac{I_2}{I_1} = \frac{r_1^2}{r_2^2}$
$I_2 = I_1~\frac{r_1^2}{r_2^2}$
$I_2 = (0.10~W/m^2)\frac{(50~m)^2}{(1000~m)^2}$
$I_2 = 2.5\times 10^{-4}~W/m^2$
The intensity at a distance of 1000 meters is $2.5\times 10^{-4}~W/m^2$
(b) We can find the maximum distance $r_3$ at which the siren can be heard.
$\frac{I_3}{I_1} = \frac{r_1^2}{r_3^2}$
$r_3^2 = \frac{I_1~r_1^2}{I_3}$
$r_3 = \sqrt{\frac{I_1}{I_3}}~r_1$
$r_3 = \sqrt{\frac{0.10~W/m^2}{1\times 10^{-6}~W/m^2}}~(50~m)$
$r_3 = 16,000~m = 16~km$
The maximum distance at which the siren can be heard is 16 km.
