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