Question

the orbital period, p, of a planet and the planets distance from the sun, a, in astronomical units is related by the formula p = a^{\frac{3}{2}}. if saturns orbital period is 29.5 years, what is its distance from the sun? 44.3 au 160.2 au 9.5 au 19.7 au

Explanation:

Step1: Given the formula

$P = a^{\frac{3}{2}}$, where $P = 29.5$.

Step2: Solve for $a$

First, rewrite the equation as $a^{\frac{3}{2}}=29.5$. Then raise both sides to the $\frac{2}{3}$ - power: $a = 29.5^{\frac{2}{3}}$.

Step3: Calculate $29.5^{\frac{2}{3}}$

$a=\sqrt[3]{29.5^{2}}=\sqrt[3]{870.25}\approx9.5$.

Answer:

$9.5$ AU