2011 AIME I Problem 13

A cube with side length 10 is suspended above a plane. The vertex closest to the plane is labeled A. The three vertices adjacent to vertex A are at heights 10, 11, and 12 above the plane. The distance from vertex A to the plane can be expressed as \frac{r-\sqrt{s}}{t}, where r, s, and t are positive integers. Find r+s+t.