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 the orbital period of planet y is twice the orbital period of planet x, by what factor is the mean distance increased?

Explanation:

Step1: Write the formula for both planets

Let the orbital - period of planet X be $T_X$ and its mean distance from the sun be $A_X$, and for planet Y be $T_Y$ and $A_Y$. The formula is $T^{2}=A^{3}$. So for planet X, $T_X^{2}=A_X^{3}$, and for planet Y, $T_Y^{2}=A_Y^{3}$.

Step2: Use the given relationship between periods

We know that $T_Y = 2T_X$. Substitute $T_Y$ into its formula: $(2T_X)^{2}=A_Y^{3}$. Expand the left - hand side: $4T_X^{2}=A_Y^{3}$.

Step3: Substitute $T_X^{2}=A_X^{3}$

Since $T_X^{2}=A_X^{3}$, we have $4A_X^{3}=A_Y^{3}$. Then $A_Y^{3}=4A_X^{3}$, and $A_Y = 4^{\frac{1}{3}}A_X=2^{\frac{2}{3}}A_X$.

Answer:

$2^{\frac{2}{3}}$