Chapter 11.6, Problem 49PPE

To determine

If you decrease your distance from the source of a sound by half, by what factor will the intensity of the sound increase? Explain.

Answer to Problem 49PPE

4

Explanation of Solution

Given information:

The intensity of a sound s varies inversely with the square of the distance d from the sound.

This can be modeled by the equation $s{d}^2=k$ where $k$ is a constant.

Concept Used:

• Mathematically relationship between two variables that can be expressed by an equation in which one variable is equal to constant times the other variable is called direct variation.
• Mathematically relationship between two variables that can be expressed by an equation in which the product of two variables is constant is called inverse variation.

Calculation:

According to question s varies inversely with the square of the distance d.

  $s{d}^2=k$

As distance from the source of a sound decrease by half

The decreased distance = $d\text{'}$

  $\begin{array}{c}d\text{'}=\frac{d}{2}\\ nowsd\text{'}=k\\ s\cdot {\left(\frac{d}{2}\right)}^2=k\\ \frac{s\cdot d^2}{4}=k\\ s\cdot d^2=4k\\ sos=\frac{4k}{d^2}\end{array}$

where k is constant and $k\ne 0$

so the intensity of the sound increases 4 times.