Question

the equation ( t^{2}=a^{3} ) shows the relationship between a planets orbital period, ( t ), and the planets 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^{\frac{1}{3}} )
( k^{\frac{1}{2}} )
( k^{\frac{2}{3}} )
( k^{\frac{3}{2}} )

Explanation:

Step1: Write the equations for both planets

Let \(T_X\) be the orbital period of planet \(X\) and \(A_X\) be its mean - distance from the sun. So, \(T_X^{2}=A_X^{3}\).
Let \(T_Y\) be the orbital period of planet \(Y\) and \(A_Y = kA_X\) be its mean - distance from the sun. Then \(T_Y^{2}=A_Y^{3}=(kA_X)^{3}=k^{3}A_X^{3}\).

Step2: Express \(T_Y\) in terms of \(T_X\)

Since \(T_X^{2}=A_X^{3}\), we can substitute \(A_X^{3}=T_X^{2}\) into the equation for \(T_Y^{2}\).
We get \(T_Y^{2}=k^{3}T_X^{2}\).
Take the square root of both sides: \(T_Y = T_Xk^{\frac{3}{2}}\).

Answer:

\(k^{\frac{3}{2}}\)