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]
