To solve the formula [tex]\( P = 4 \pi I r^2 \)[/tex] for the distance [tex]\( r \)[/tex], we will follow these steps rigorously:
1. Start with the given formula:
[tex]\[
P = 4 \pi I r^2
\][/tex]
2. Isolate the term containing [tex]\( r \)[/tex]:
[tex]\[
r^2 = \frac{P}{4 \pi I}
\][/tex]
3. Solve for [tex]\( r \)[/tex]:
[tex]\[
r = \pm \sqrt{\frac{P}{4 \pi I}}
\][/tex]
Since the distance [tex]\( r \)[/tex] cannot be negative, we take the positive solution. Thus, we have:
[tex]\[
r = \sqrt{\frac{P}{4 \pi I}}
\][/tex]
4. Simplify the expression further. Notice that:
[tex]\[
\sqrt{\frac{P}{4 \pi I}} = \frac{1}{2} \sqrt{\frac{P}{\pi I}}
\][/tex]
Thus, the simplified formula for [tex]\( r \)[/tex] is:
[tex]\[
r = \frac{1}{2} \sqrt{\frac{P}{\pi I}}
\][/tex]
So, the correct answer is:
[tex]\[
\boxed{C. \, r = \frac{1}{2} \sqrt{\frac{P}{\pi I}}}
\][/tex]
