Answer
(a) The intensity when the wave passed 1.0 km from the source was $8.7\times 10^9~J/m^2\cdot s$
(b) Energy passed through an area of $2.0~m^2$ at a rate of $1.7\times 10^{10}~W$
Work Step by Step
(a) Let $I_1$ be the intensity at a distance of 1.0 km from the source. Let $I_2$ be the intensity at a distance of 54.0 km from the source.
$\frac{I_1}{I_2} = \frac{r_2^2}{r_1^2}$
$I_1 = (\frac{r_2^2}{r_1^2})~I_2$
$I_1 = \frac{(5.4\times 10^4~m)^2}{(1.0\times 10^3~m)^2}~(3.0\times 10^6~J/m^2\cdot s)$
$I_1 = 8.7\times 10^9~J/m^2\cdot s$
The intensity when the wave passed 1.0 km from the source was $8.7\times 10^9~J/m^2\cdot s$
(b) $I = \frac{power}{area}$
$P = I~A$
$P = (8.7\times 10^9~J/m^2\cdot s)~(2.0~m^2)$
$P = 1.7\times 10^{10}~W$
Energy passed through an area of $2.0~m^2$ at a rate of $1.7\times 10^{10}~W$.
