Question

multiple choice 17 points
keplers third law states $p = a^{\frac{3}{2}}$ where:
p = the time it takes the planet to go around the sun, in years.
a = the planets distance from the sun, in astronomical units.
if the orbital period of jupiter is 9.3 years, what is its distance from the sun to the nearest hundredth of an astronomical unit?
4.42 units
5.43 units
5.42 units
4.91 units

Explanation:

Step1: Rearrange Kepler's third law

Given $P = a^{\frac{3}{2}}$, we can solve for $a$. Raise both sides to the $\frac{2}{3}$ power, so $a=P^{\frac{2}{3}}$.

Step2: Substitute the value of $P$

We know $P = 9.3$. Then $a=(9.3)^{\frac{2}{3}}$.
Using a calculator, $(9.3)^{\frac{2}{3}}=\sqrt[3]{9.3^{2}}=\sqrt[3]{86.49}\approx4.42$.

Answer:

A. 4.42 units