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 13.2 years, what is its distance from the sun to the nearest hundredth of an astronomical unit?
5.31 units
5.59 units
6.21 units
56 units

Explanation:

Step1: Rearrange the formula

Given $P = a^{\frac{3}{2}}$, we can solve for $a$ by raising both sides to the $\frac{2}{3}$ - power. So $a=P^{\frac{2}{3}}$.

Step2: Substitute the value of $P$

Substitute $P = 13.2$ into the formula $a = P^{\frac{2}{3}}$. Then $a=(13.2)^{\frac{2}{3}}=\sqrt[3]{13.2^{2}}=\sqrt[3]{174.24}$.

Step3: Calculate the value

Using a calculator, $\sqrt[3]{174.24}\approx5.59$.

Answer:

B. 5.59 units