Question

A listener at a distance from the source hears a sound with intensity of 7.14 x 10^-7 W/m².  

When the listener moved farther from the source to a distance of 132.5 m, he hears an intensity of 2.94 x 10^-7 W/m². What is its initial distance from the source of sound?

Answer 1

The intensity of the sound is inversely proportional to the distance from it. Thus:

$$I_1 \propto\dfrac{1}{r_1^2}\\ I_2 \propto\dfrac{1}{r_2^2}$$

where $I_1 =7.14 \times 10^{-7}W/m^2 , I_2=2.94 \times 10^{-7}W/m^2$ are the initial and the final intensity respectively, $r_1,r_2 = 132.5m$ are the initial and the final distances. Dividing the second equation by the first one, and expressing $r_1$, obtain:

$$\dfrac{I_2}{I_1} = \dfrac{r_1^2}{r_2^2} \\ r_1 = r_2\sqrt{\dfrac{I_2}{I_1}}\\ r_1 = 132.5\cdot \sqrt{\dfrac{2.94\times 10^{-7}}{7.14\times 10^{-7}}}\approx 85m$$

Answer. 85 m.