In this question, we are given the radius of the orbits of two planets around the Sun. Let`s call the radius of the first planet`s orbit R1 and the radius of the second planet`s orbit R2.

We are asked to find the ratio of their periods, T1/T2.

According to Kepler`s third law of planetary motion, the square of the period of a planet (T) is directly proportional to the cube of its average distance from the Sun (r^3). Mathematically, this can be represented as T^2 ∝ r^3.

Since both planets are orbiting the same star (the Sun), we can use this law to find the ratio of their periods.

Let`s write down the equation for both planets:

For the first planet, T1^2 ∝ R1^3

For the second planet, T2^2 ∝ R2^3

Now, let`s substitute the given values for R1 and R2:

For the first planet, T1^2 ∝ R^3

For the second planet, T2^2 ∝ (4R)^3

Simplifying the equation for the second planet, we get:

T2^2