Question

a planet takes 8 years to orbit the sun. if another planet is twice as far from the sun, how long does it take to orbit the sun?
a 32 years
b 11.3 years
c 22.6 years
d 16 years

Explanation:

Step1: Recall Kepler's third law

$T^{2}\propto r^{3}$, where $T$ is the orbital - period and $r$ is the semi - major axis of the orbit. Let $T_1$ and $r_1$ be the period and semi - major axis of the first planet, and $T_2$ and $r_2$ be those of the second planet. We know that $T_1 = 8$ years and $r_2=2r_1$.

Step2: Set up the ratio

From $T^{2}\propto r^{3}$, we have $\frac{T_{2}^{2}}{T_{1}^{2}}=\frac{r_{2}^{3}}{r_{1}^{3}}$. Substitute $r_2 = 2r_1$ into the equation: $\frac{T_{2}^{2}}{T_{1}^{2}}=\frac{(2r_1)^{3}}{r_{1}^{3}}=\frac{8r_{1}^{3}}{r_{1}^{3}} = 8$.

Step3: Solve for $T_2$

Since $T_1 = 8$ years and $\frac{T_{2}^{2}}{T_{1}^{2}}=8$, then $T_{2}^{2}=8T_{1}^{2}$. Substitute $T_1 = 8$ into the equation: $T_{2}^{2}=8\times8^{2}=8^{3}=512$. So, $T_2=\sqrt{512}\approx22.6$ years.

Answer:

C. 22.6 years