the orbital period, $p$, of a planet and the planets distance from the sun, $a$, in astronomical units is…

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? 9.5 au 160.2 au 44.3 au 19.7 au

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? 9.5 au 160.2 au 44.3 au 19.7 au

Answer

Explanation:

Step1: Given the formula and values

We have $P = a^{\frac{3}{2}}$ and $P = 29.5$. So, $29.5=a^{\frac{3}{2}}$.

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: $a = 29.5^{\frac{2}{3}}$. Using a calculator, $a=29.5^{\frac{2}{3}}\approx9.5$.

Answer:

9.5 AU