Problem: What is the largest possible distance between two points, one on the sphere of radius 19 with center $(-2,-10,5),$ and the other on the sphere of radius 87 with center $(12,8,-16)$? Solution: It is not hard to see that the ball with radius $19$ is entirely contained inside the sphere of radius $87$. Thus, the answer is simply the distance between the two centers plus $19+87$ assuming we are optimally extending the line between the centers as far as we can in either direction. The answer is thus $19+87+\sqrt{14^2+18^2+21^2} = 137.$