Answer:
Given the equation:
[tex]T^2 =A^3[/tex]
shows the relationship between a planet's orbital period T and the planet's mean distance from the sun, A in Astronomical units
then:
For planet X:
Orbital period is:
[tex]T_{X} = (A)^{\frac{3}{2}}[/tex] .....[1]
As per the statement:
If planet Y is k times the mean distance from the sun as planet X.
⇒ A planet Y= kA mean distance from the sun as planet X.
then orbital period of Planet Y is:
[tex]T_{Y} = (kA)^{\frac{3}{2}}=k^{\frac{3}{2}}\cdot (A)^{\frac{3}{2}}[/tex] ....[2]
Divide equation [2] by [1] we have;
[tex]\frac{T_{Y}}{T_{X}} = \frac{k^{\frac{3}{2}}\cdot (A)^{\frac{3}{2}}}{A^{\frac{3}{2}}}[/tex]
Simplify:
[tex]\frac{T_{Y}}{T_{X}} =k^{\frac{3}{2}}[/tex]
or
[tex]T_{Y} =k^\frac{3}{2} T_X[/tex]
Therefore, the orbital period is increased by factor [tex]k^\frac{3}{2}[/tex]
