Three mutually tangent spheres of radius 1 rest on a horizontal plane. A sphere of radius 2 rests on them. What is the distance from the plane to the top of the larger sphere?
+130071 If we connect the three radii of the smaller circles, we will have an equilateral triangle of side 2
And connecting each of these to the center of the largest sphere will create a triangular pyramid with side edges of 3 = the distance between the centers of a smaller and larger sphere
The distance from any bottom vertex of the triangle to its center is given by :
1 / cos 30° = 2 / √3 units
The height of this pyramid is given by
√ [3^2 - ( 2/√3)^2] = √ [ 9 - 4/3] = √[ 23/3] units
So.....the height of the top of the larger sphere from the plane is given by
vertical distance from the plane to the center of a smaller sphere + height of the pyramid + radius of larger sphere =
1 + √[23/3] + 2 =
3 + √[23/3] units ≈ 5.77 units
