Question

a comet has an orbit size of 3.5 au & its orbit time is 6.54 years. t² = 42.875 years. what does r³ equal?
42.875 au
3.5 au
6.54 au
12.25 au

Explanation:

Step1: Recall Kepler's Third Law

Kepler's Third Law states that \(T^{2}=R^{3}\) for a celestial body in orbit around the Sun, where \(T\) is the orbital period (in years) and \(R\) is the semi - major axis of the orbit (in AU).

Step2: Substitute the given value of \(R\)

Given \(R = 3.5\) AU. Then \(R^{3}=(3.5)^{3}\).
We know that \((3.5)^{3}=3.5\times3.5\times3.5=(3 + 0.5)\times(3 + 0.5)\times(3+0.5)\).
First, \((3 + 0.5)\times(3 + 0.5)=3^{2}+2\times3\times0.5 + 0.5^{2}=9 + 3+0.25 = 12.25\).
Then \(12.25\times3.5=(12+0.25)\times3.5=12\times3.5+0.25\times3.5 = 42+0.875=42.875\).

Answer:

42.875 AU