Question

the orbital period, p, of a planet and the planets distance from the sun, a in astronomical units is related by the formula. if saturns orbital period is 29.5 years, what is its distance from the sun? 9.5 au 10.7 au 100.2 au

Explanation:

1. First, recall Kepler's third - law formula:

• The formula relating the orbital period \(P\) (in years) of a planet and its average distance \(a\) (in astronomical units, AU) from the Sun is \(P^{2}=a^{3}\).

1. Then, substitute the given value of \(P\) into the formula:

• We are given that \(P = 29.5\) years. Substituting \(P\) into the formula \(P^{2}=a^{3}\), we get \(a^{3}=P^{2}=(29.5)^{2}\).
• Calculate \((29.5)^{2}\):
• \((29.5)^{2}=29.5\times29.5 = 870.25\).

1. Next, solve for \(a\):

• To find \(a\), we take the cube - root of both sides of the equation \(a^{3}=870.25\). So, \(a=\sqrt[3]{870.25}\).
• Using a calculator, \(\sqrt[3]{870.25}\approx9.5\) AU.

Answer:

A. 9.5 AU