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The formula to find the period of orbit of a satellite around a planet is [tex]T^2=\left(\frac{4 \pi^2}{G M}\right) r^3[/tex], where [tex]r[/tex] is the orbit's mean radius, [tex]M[/tex] is the mass of the planet, and [tex]G[/tex] is the universal gravitational constant. If you are given all the values except [tex]r[/tex], how do you rewrite the formula to solve for [tex]r[/tex]?

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GinnyAnswer
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]
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