Question

the loudness of a sound is inversely proportional to the square of your distance from the source of the sound. if your friend is right next to the speakers at a loud concert and you are four times as far away from the speakers, how does the loudness of the music at your position compare to the loudness at your friend’s position?
the sound is 1/16 as loud at your position.
the sound is 1/4 as loud at your position.
the sound is equally loud at your position.
the sound is 4 times as loud at your position.
the sound is 16 times as loud at your position.

Explanation:

Step1: Define inverse square relation

Let loudness be $L$, distance be $d$. The relation is $L = \frac{k}{d^2}$ where $k$ is a constant.

Step2: Assign distances

Let friend's distance $d_1 = d$, your distance $d_2 = 4d$.

Step3: Calculate friend's loudness

$L_1 = \frac{k}{d_1^2} = \frac{k}{d^2}$

Step4: Calculate your loudness

$L_2 = \frac{k}{d_2^2} = \frac{k}{(4d)^2} = \frac{k}{16d^2}$

Step5: Compare loudness

$\frac{L_2}{L_1} = \frac{\frac{k}{16d^2}}{\frac{k}{d^2}} = \frac{1}{16}$

Answer:

The sound is 1/16 as loud at your position.