To solve this problem we will apply the concepts related to the Area, the power and the proportionality relationships between intensity and distance.
The expression for sound power is,
![P = AI](https://tex.z-dn.net/?f=P%20%3D%20AI)
Here,
A = Area
I = Intensity
P = Power
At the same time the area can be written as,
![A = \frac{\pi d^2}{4}](https://tex.z-dn.net/?f=A%20%3D%20%5Cfrac%7B%5Cpi%20d%5E2%7D%7B4%7D)
Now the intensity is inversely proportional to the square of the distance from the source, then
![I \propto \frac{1}{r^2}](https://tex.z-dn.net/?f=I%20%5Cpropto%20%5Cfrac%7B1%7D%7Br%5E2%7D)
The expression for the intensity at different distance is
![\frac{I_1}{I_2}= \frac{r^2_2}{r_1^2}](https://tex.z-dn.net/?f=%5Cfrac%7BI_1%7D%7BI_2%7D%3D%20%5Cfrac%7Br%5E2_2%7D%7Br_1%5E2%7D)
Here,
= Intensity at distance 1
= Intensity at distance 2
= Distance 1 from light source
= Distance 2 from the light source
If we rearrange the expression to find the intensity at second position we have,
![I_2 = I_1 (\frac{r_1^2}{r_2^2})](https://tex.z-dn.net/?f=I_2%20%3D%20I_1%20%28%5Cfrac%7Br_1%5E2%7D%7Br_2%5E2%7D%29)
If we replace with our values at this equation we have,
![I_2 = (0.10W/m^2)(\frac{1.0m^2}{30.0m^2})](https://tex.z-dn.net/?f=I_2%20%3D%20%280.10W%2Fm%5E2%29%28%5Cfrac%7B1.0m%5E2%7D%7B30.0m%5E2%7D%29)
![I_2 = 1.11*10^{-4} W/m^2](https://tex.z-dn.net/?f=I_2%20%3D%201.11%2A10%5E%7B-4%7D%20W%2Fm%5E2)
Now using the equation to find the area we have that
![A = \frac{\pi (8.4*10^{-3}m)^2}{4}](https://tex.z-dn.net/?f=A%20%3D%20%5Cfrac%7B%5Cpi%20%288.4%2A10%5E%7B-3%7Dm%29%5E2%7D%7B4%7D)
![A = 5.5*10^{-5}m^2](https://tex.z-dn.net/?f=A%20%3D%205.5%2A10%5E%7B-5%7Dm%5E2)
Finally with the intensity and the area we can find the sound power, which is
![P = AI](https://tex.z-dn.net/?f=P%20%3D%20AI)
![P = (5.5*10^{-5}m^2)(1.11*10^{-4}W/m^2)](https://tex.z-dn.net/?f=P%20%3D%20%285.5%2A10%5E%7B-5%7Dm%5E2%29%281.11%2A10%5E%7B-4%7DW%2Fm%5E2%29)
![P = 6.1*10^{-9}J/s](https://tex.z-dn.net/?f=P%20%3D%206.1%2A10%5E%7B-9%7DJ%2Fs)
Power is defined as the quantity of Energy per second, then
![E = 6.1*10^{-9}J](https://tex.z-dn.net/?f=E%20%3D%206.1%2A10%5E%7B-9%7DJ)