Question

The equation t^3=a^2 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 the orbital period of planet Y is twice the orbital period of planet X, by what factor is the mean distance increased?

Answer 1

Answer:

     2√2

Step-by-step explanation:

We can find the relationship of interest by solving the given equation for A, the mean distance.

Solve for A

  $T^3=A^2\\\\A=\sqrt{T^3}=T\sqrt{T}\qquad\text{take the square root}$

Substitute values

The mean distance of planet X is found in terms of its period to be ...

  $D_x=T_x\sqrt{T_x}$

The mean distance of planet Y can be found using the given relation ...

  $T_y=2T_x\\\\D_y=T_y\sqrt{T_y}=2T_x\sqrt{2T_x}=(2\sqrt{2})T_x\sqrt{T_x}\\\\D_y=2\sqrt{2}\cdot D_x$

The mean distance of planet Y is increased from that of planet X by the factor ...

  2√2