Answered

Sound waves can be ranked by their intensity, [tex]I[/tex], given in the formula below, where [tex]r[/tex] is the distance from the source of a sound with a power output of [tex]P[/tex].

[tex]\[ P = 4 \pi I r^2 \][/tex]

Solve the formula for the distance from the source, [tex]r[/tex].

A. [tex] r = \frac{1}{4} \sqrt{\frac{\pi I}{P}} [/tex]
B. [tex] r = \frac{1}{2} \sqrt{\frac{\pi I}{P}} [/tex]
C. [tex] r = \frac{1}{4} \sqrt{\frac{P}{\pi I}} [/tex]
D. [tex] r = \frac{1}{2} \sqrt{\frac{P}{\pi I}} [/tex]



Answer :

GinnyAnswer
Let's solve the given formula for the distance [tex]\( r \)[/tex].

We start with the given formula:
[tex]\[ P = 4 \pi I r^2 \][/tex]

Our goal is to isolate [tex]\( r \)[/tex] on one side of the equation. Here are the steps:

1. Divide both sides of the equation by [tex]\( 4 \pi I \)[/tex] to isolate [tex]\( r^2 \)[/tex]:
[tex]\[ r^2 = \frac{P}{4 \pi I} \][/tex]

2. Take the square root of both sides to solve for [tex]\( r \)[/tex]:
[tex]\[ r = \sqrt{\frac{P}{4 \pi I}} \][/tex]

3. Simplify the expression under the square root:
[tex]\[ r = \sqrt{\frac{P}{4 \pi I}} = \sqrt{\frac{P}{4} \cdot \frac{1}{\pi I}} = \frac{1}{2} \sqrt{\frac{P}{\pi I}} \][/tex]

So, the correct expression for [tex]\( r \)[/tex] is:
[tex]\[ r = \frac{1}{2} \sqrt{\frac{P}{\pi I}} \][/tex]

Out of the given choices, this corresponds to:
D. [tex]\( r = \frac{1}{2} \sqrt{\frac{P}{\pi I}} \)[/tex]