We discover a nearby star with two planets. the first planet has an orbit period of 10 years and is in a circular orbit with radius 106 km. the second planet has an orbit period of 15 years. what is its orbit radius? you may assume it is also in a circular orbit.

139 Km.

Explanation:

The question tells us that a planet A has an orbit period of 10 years and its circular orbit has a radius of 106 Km, whilst a planet B has an orbit period of 15 years (also assuming a circular orbit), both orbiting a nearby star.

This information allow us to use the Kepler's 3rd law, for the special case in which the orbit is circular.

Kepler's 3rd law, tells that there exist a direct proportionality between the square of the orbit period, and the cube of the orbit radius (in the more general case, with the cube of the semi-major axis of the elipse), for celestial bodies orbiting a same star.

(like Earth and Mars orbiting Sun).

So, for planet A and planet B (orbiting a same star), we can write the following:

(TA)²/ (TB)² = (rA)³ / (rB)³

Replacing by TA= 10 years, TB= 15 years, rA= 106 Km, and solving for Rb, we get  RB= 139 Km.