Question

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?
options:
( 2^{\frac{1}{3}} )
( 2^{\frac{1}{2}} )
( 2^{\frac{2}{3}} )
( 2^{\frac{3}{2}} )

Explanation:

Step1: Define Variables for Planets

Let the orbital period of planet X be \( T_X \) and its mean distance from the sun be \( A_X \). For planet Y, the mean distance \( A_Y = 2A_X \), and let its orbital period be \( T_Y \).

Step2: Apply the Given Equation

The equation is \( T^2 = A^3 \). For planet X: \( T_X^2 = A_X^3 \). For planet Y: \( T_Y^2 = A_Y^3=(2A_X)^3 = 8A_X^3 \).

Step3: Relate \( T_Y \) and \( T_X \)

From planet X's equation, \( A_X^3=T_X^2 \). Substitute into planet Y's equation: \( T_Y^2 = 8T_X^2 \). Take square roots: \( T_Y=\sqrt{8}T_X = 2^{\frac{3}{2}}T_X \). So the factor is \( 2^{\frac{3}{2}} \).

Answer:

\( 2^{\frac{3}{2}} \) (the last option, \( 2^{\frac{3}{2}} \))