Question

according to keplers third law (p² = a³), a planet with an orbital period of 12 years lies at an average distance from the sun equal to ______ astronomical units (au).
question 8
1 pts
according to keplers third law (p² = a³), a planet with a semimajor axis of 9.58 au has an average orbital period of ______ years.
question 9
1 pts
the planets in the previous two questions are both a select distance from the sun than earth is, therefore, according to keplers third law, they both take a select time to orbit the sun.

Explanation:

Question 1

Step1: Given Kepler's third law $p^{2}=a^{3}$, where $p = 12$ years. We need to find $a$.

First, substitute $p$ into the formula: $12^{2}=a^{3}$, so $a^{3}=144$.

Step2: Solve for $a$.

$a=\sqrt[3]{144}\approx5.24$ AU

Step1: Given Kepler's third law $p^{2}=a^{3}$, where $a = 9.58$ AU. We need to find $p$.

Substitute $a$ into the formula: $p^{2}=9.58^{3}$. Calculate $9.58^{3}=9.58\times9.58\times9.58 = 879.43$.

Step2: Solve for $p$.

$p=\sqrt{879.43}\approx29.65$ years

The first planet has a distance of about $5.24$ AU and the second has $9.58$ AU. Earth has an average distance of 1 AU from the Sun. So both planets are at a greater distance from the Sun than Earth. According to Kepler's third law, the farther a planet is from the Sun (larger $a$), the longer its orbital - period ($p$).

Answer:

$5.24$

Question 2