the equation ( t^{2}=a^{3} ) shows the relationship between a planets orbital period, ( t ), and the planets…

Explanation:

Step1: Substitute the orbital periods into the formula

Let the orbital period of planet (X) be (T_{X}), and its mean distance from the sun be (A_{X}). So (T_{X}^{2}=A_{X}^{3}). Let the orbital period of planet (Y) be (T_{Y}), and its mean distance from the sun be (A_{Y}). Given (T_{Y} = 2T_{X}), then ((2T_{X})^{2}=A_{Y}^{3}).

Step2: Express (A_{Y}) in terms of (A_{X})

Since (T_{X}^{2}=A_{X}^{3}), substitute (T_{X}^{2}) into ((2T_{X})^{2}=A_{Y}^{3}). We have (4T_{X}^{2}=A_{Y}^{3}), and because (T_{X}^{2}=A_{X}^{3}), then (4A_{X}^{3}=A_{Y}^{3}). Take the cube - root of both sides: (A_{Y}=\sqrt[3]{4}A_{X}\approx1.59A_{X}).

Answer:

The mean distance of planet (Y) from the sun is approximately (1.59) times the mean distance of planet (X) from the sun.