Chapter 5.3, Problem 7MRPS

Solution

To calculate: The intensity of the sound we hear if we are $15$ meters from the stage.

The intensity of the sound we hear if we are $15$ meters from the stage is $0.04W/m^2$.

Given information: The intensity $I$ of a sound (in watts per square meter) varies inversely with the square of the distance $d$ (in meters) from the source of the sound. At a distance of $1$ meter from the stage, the sound intensity of a rock concert is about $10$ watts per square meter.

Calculation:

The general equation for inverse variation is-

  $y=\frac{a}{x},a\ne 0$

Given, The intensity $I$ of a sound varies inversely with the square of the distance $d$ from the source of the sound.

  $\Rightarrow I=\frac{a}{d^2}$ , $a\ne 0$ ……(1)

At a distance of $1m$ from the stage, the sound intensity of a rock concert is about $10W/m^2$.

  $\Rightarrow$$I=10W/m^2$ when $d=1m$.

Substitute $1$ for $d$ and $10$ for $I$ in (1).

  $\begin{array}{l}10=\frac{a}{{(1)}^2}\\ a=10\end{array}$

Thus, $I=\frac{10}{d^2}$.

Substitute $15$ for $d$ to find the intensity.

  $\begin{array}{l}I=\frac{10}{{(15)}^2}\\ I=\frac{10}{225}\\ I=0.04\end{array}$

Thus, The intensity of the sound we hear if we are $15$ meters away from the stage is $0.04W/m^2$.