Question

The equation T squared = A cubed 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 k times the mean distance from the sun as planet X, by what factor is the orbital period increased?
k Superscript one-third
k Superscript one-half
k Superscript two-thirds
k Superscript three-halves

Answer 1

Use Kepler's law: T^2 = A^3, so T = A^{3/2}. If A_Y = k A_X, then
T_Y = (k A_X)^{3/2} = k^{3/2} T_X.
So the period increases by a factor of k^{3/2} (k superscript three-halves).