Question

The equation mc024-1.jpg 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?
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Answer 1

Given that 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.
i.e. $T= \sqrt{A^3}$

If planet Y is twice the mean distance from the sun as planet X,
i.e. $A_Y=2A_X$,
then,
$\frac{T_Y}{T_X} = \frac{\sqrt{(2A_X)^3}}{\sqrt{(A_X)^3}} = \frac{(2A_X)^3}{(A_X)^3} = \frac{8(A_X)^3}{(A_X)^3} =8$

Therefore, the orbital period increased by a factor of 2^3/2