Question

1. suppose a new asteroid was recently discovered which takes 557 months to orbit the sun once (thats equal to 16,700 days or 46.4 years.). what is its average distance from the sun? (show your work.)

Explanation:

Step1: Recall Kepler's third - law

Kepler's third - law is $T^{2}=kR^{3}$, where $T$ is the orbital period, $R$ is the average distance from the central body, and $k$ is a constant. For objects orbiting the Sun, when $T$ is in years and $R$ is in astronomical units (AU), $k = 1$.

Step2: Rearrange the formula for $R$

From $T^{2}=R^{3}$, we can solve for $R$ as $R = T^{\frac{2}{3}}$.

Step3: Substitute the given value of $T$

The orbital period $T = 46.4$ years. Then $R=(46.4)^{\frac{2}{3}}$.
First, calculate $46.4^{2}=2152.96$. Then find the cube - root of $2152.96$. $\sqrt[3]{2152.96}\approx12.9$.

Answer:

The average distance of the asteroid from the Sun is approximately $12.9$ AU.