Let's start with the given formula:
[tex]\[ P = 4 \pi I r^2 \][/tex]
We need to solve for [tex]\( r \)[/tex]. To do this, we will isolate [tex]\( r \)[/tex] on one side of the equation. Here are the steps:
1. Divide both sides by [tex]\( 4 \pi I \)[/tex] to isolate [tex]\( r^2 \)[/tex]:
[tex]\[ \frac{P}{4 \pi I} = r^2 \][/tex]
2. Take the square root of both sides to solve for [tex]\( r \)[/tex]:
[tex]\[ r = \sqrt{\frac{P}{4 \pi I}} \][/tex]
Notice that [tex]\( \sqrt{\frac{P}{4 \pi I}} \)[/tex] can be rewritten by factoring out the constants under the square root:
[tex]\[ r = \sqrt{\frac{P}{4 \pi I}} = \sqrt{\frac{P}{4 \pi I}} = \frac{\sqrt{P}}{\sqrt{4 \pi I}} \][/tex]
Since [tex]\(\sqrt{4} = 2\)[/tex], we have:
[tex]\[ r = \frac{\sqrt{P}}{2 \sqrt{\pi I}} = \frac{1}{2} \sqrt{\frac{P}{\pi I}} \][/tex]
Therefore, the correct answer is:
A. [tex]\( r = \frac{1}{2} \sqrt{\frac{P}{\pi I}} \)[/tex]
