To solve for the mean radius [tex]\(r\)[/tex] of the orbit in the formula [tex]\( T^2 = \left(\frac{4 \pi^2}{G M}\right) r^3 \)[/tex], we need to isolate [tex]\(r\)[/tex]. Here are the steps:
1. Start with the given formula:
[tex]\[
T^2 = \left(\frac{4 \pi^2}{G M}\right) r^3
\][/tex]
2. To isolate [tex]\( r^3 \)[/tex], multiply both sides of the equation by [tex]\(\frac{G M}{4 \pi^2}\)[/tex]:
[tex]\[
T^2 \cdot \frac{G M}{4 \pi^2} = r^3
\][/tex]
3. Simplify the equation:
[tex]\[
r^3 = \frac{T^2 G M}{4 \pi^2}
\][/tex]
4. To solve for [tex]\( r \)[/tex], take the cube root of both sides:
[tex]\[
r = \left( \frac{T^2 G M}{4 \pi^2} \right)^{1/3}
\][/tex]
So, the formula to solve for the mean radius [tex]\( r \)[/tex] is:
[tex]\[
r = \left( \frac{T^2 G M}{4 \pi^2} \right)^{1/3}
\][/tex]
