Question

The equation T^2 = A^3 shows the relationship between a planet’s orbital period, T, and the planet’s mean distance from the sun, A, in astronomical units, AU. If planet Y is twice the mean distance from the sun as planet X, by what factor is the orbital period increased?

Answer 1

So, let planet X's orbital period be T and its mean distance from the sun be A. Also let planet Y's orbital period be T_1, so that means if planet Y's mean distance from the sun were double that of planet X:

$Z^2=(2A)^3\\ Z^2=8A^3\implies\\ Z^2=8T^2 \implies\\ Z=2\sqrt{2}T$

Which means that the orbital period in planet Y is increased by a factor of $2\sqrt{2}$