the equation $t^{2}=a^{3}$ shows the relationship between a planets orbital period, $t$, and the planets…

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 twice the mean distance from the sun as planet x, by what factor is the orbital period increased?\n$2^{\frac{1}{3}}$\n$2^{\frac{1}{2}}$\n$2^{\frac{2}{3}}$\n$2^{\frac{3}{2}}$

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 twice the mean distance from the sun as planet x, by what factor is the orbital period increased?\n$2^{\frac{1}{3}}$\n$2^{\frac{1}{2}}$\n$2^{\frac{2}{3}}$\n$2^{\frac{3}{2}}$

Answer

Explanation:

Step1: Define variables for planets X and Y

Let the mean - distance of planet X from the sun be $A_X$ and its orbital period be $T_X$, so $T_X^{2}=A_X^{3}$. Let the mean - distance of planet Y from the sun be $A_Y = 2A_X$ and its orbital period be $T_Y$, so $T_Y^{2}=A_Y^{3}$.

Step2: Substitute $A_Y = 2A_X$ into the equation for planet Y

Since $T_Y^{2}=A_Y^{3}$ and $A_Y = 2A_X$, we have $T_Y^{2}=(2A_X)^{3}=8A_X^{3}$.

Step3: Express $T_Y$ in terms of $T_X$

We know that $T_X^{2}=A_X^{3}$, so $A_X^{3}=T_X^{2}$. Substituting $A_X^{3}=T_X^{2}$ into $T_Y^{2}=8A_X^{3}$, we get $T_Y^{2}=8T_X^{2}$. Then $T_Y=\sqrt{8}T_X = 2^{\frac{3}{2}}T_X$.

Answer:

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