We are given that a comet has a period of 84 years. We are asked to determine the length of its semi-major axis. To do that we will use the following formula:
[tex]p^2=a^3[/tex]This is Kepler's third law, where:
[tex]\begin{gathered} p=\text{period} \\ a=\text{ distance to semi-major ax}is \end{gathered}[/tex]Now, we solve for "a" by taking the cubic root to both sides:
[tex]\sqrt[3]{p^2}=a[/tex]Now we substitute the value of 84 years:
[tex]\sqrt[3]{(84)^2}=a[/tex]Solving the operations:
[tex]19.2AU=a[/tex]Therefore, the length of the semi-major axis is 19.2 Astronomical Units.
