What is the largest possible [distance](https://artofproblemsolving.com/wiki/index.php/Distance) between two [points](https://artofproblemsolving.com/wiki/index.php/Point), one on the [sphere](https://artofproblemsolving.com/wiki/index.php/Sphere) of [radius](https://artofproblemsolving.com/wiki/index.php/Radius) 19 with [center](https://artofproblemsolving.com/wiki/index.php/Center) $(-2,-10,5)$ and the other on the sphere of radius 87 with center $(12,8,-16)$?
The distance between the two centers of the spheres can be determined via the [distance formula](https://artofproblemsolving.com/wiki/index.php/Distance_formula) in three dimensions: $\sqrt{(12 - (-2))^2 + (8 - (-10))^2 + (-16 - 5)^2} = \sqrt{14^2 + 18^2 + 21^2} = 31$. The largest possible distance would be the sum of the two radii and the distance between the two centers, making it $19 + 87 + 31 = \boxed{137}$.