2021 Fall AMC 10A Problems/Problem 22
Problem
Inside a right circular cone with base radius and height are three congruent spheres with radius . Each sphere is tangent to the other two spheres and also tangent to the base and side of the cone. What is ?
Solution 1 (Coordinates)
We will use coordinates. WLOG, let the coordinates of the center of the base of the cone be the origin. Then, let the center of one of the spheres be . Note that the distance between this point and the plane given by is . Thus, by the point-to-plane distance formula, we have
Solving for yields .
~ Leo.Euler