Question

The volume, Q, of a sound is inversely proportional to the square of the distance n, in meters, from the source of the sound. A sound is heard at 75 decibels at a distance of 22 meters from its source. Write a formula for Q as a function of n, including finding the value of any unknown constants. Q = _______

Answer 1

Step 1. Define the variables for the formula.

Q --> Volume of the sound

n --> Distance in meters from the source of the sound

Step 1. Since Q is inversely proportional to the square of the distance n, the formula we will use for inverse proportionality is:

$Q=\frac{a}{n^2}$

Where a is a constant of proportionality.

Step 2. Find the value of constant a.

We are given the following known values:

''A sound is heard at 75 decibels at a distance of 22 meters from its source''

here, the values for Q and n are:

$\begin{gathered} Q=75 \\ n=22 \end{gathered}$

Ans we need to substitute them into our inverse proportionality formula in order to find a:

$\begin{gathered} Q=\frac{a}{n^2} \\ 75=\frac{a}{22^2} \end{gathered}$

Solving for a:

$\begin{gathered} 75\times22^2=a \\ 75\times484=a \\ 36,300=a \end{gathered}$

The value of the constant a is 36,300.

Step 3. Substitute the value of the constant into the formula:

$\begin{gathered} Q=\frac{a}{n^2} \\ Q=\frac{36300}{n^2} \end{gathered}$

This is the formula for Q as a function of n.

Answer:

$Q=\frac{36300}{n^2}$