Planet X orbits the star Omega with a “year” that is 200 earth days long. Planet Y circles Omega at four times the distance of planet X. How long is a year on planet Y?
This is a question on Kepler’s Third Law
Let Px be the period of revolution (that is, a ‘year’) of Planet X, Rx be the distance of Planet X, Py be the period of revolution of Planet Y and Ry be the distance of Planet Y.
Kepler’s Third Law tells us
Py²/Px² = Ry³/Rx³
Or, Py² = Px²Ry³/Rx³
You’re given that Px = 200 days. You also know the distance of planey Y is 4 times the distance of planet X, so Ry = 4Rx. Substituting this information in gives
Py² = (200 days)² (4Rx)³/Rx³
Evaluating the right hand side results in
Py² = 40,000 days² * 64 Rx³/Rx³. Simplifying and cancelling the Rx³’s:
Py² = 2,560,000 days².
Taking the square root of both sides,
Py = 1600 days. So a year on Planet Y is 1600 earth days.