Question

A listener initially 150 m from the source hears the sound with intensity I₁. When the listener moved closer to the source half of its initial distance, how many times the sound intensity he hear increases?

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 $I_1, I_2$ are the initial and the final intensity respectively, $r_1,r_2$ are the initial and the final distances. Thus, if $r_1 = 2r_2$, the ration of intensities is:

$$\dfrac{I_2}{I_1} = \dfrac{r_1^2}{r_2^2} = \dfrac{(2r_2)^2}{r_2^2} = 4$$

The sound intensity increases 4 times.

Answer. 4.