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? A. 2^1/3. B. 2^1/2. C. 2^2/3. D. 2^3/2

The given equation for the relationship between a planet's orbital period, T and the planet's mean distance from the sun, A is T^2 = A^3. Let the orbital period of planet X be T (X) and that of planet Y = T (Y) and let the mean distance of planet X from the sun be A (X) and that of planet Y = A (Y), then A (Y) = 2A (X) [T (Y) ]^2 = [A (Y) ]^3 = [2A (X) ]^3 But [T (X) ]^2 = [A (X) ]^3 Thus [T (Y) ]^2 = 2^3[T (X) ]^2 [T (Y) ]^2 / [T (X) ]^2 = 2^3 T (Y) / T (X) = 2^3/2 Therefore, the orbital period increased by a factor of 2^3/2