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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?

 Dec 24, 2018
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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

 

 

cool cool cool

 Dec 24, 2018
edited by CPhill  Dec 24, 2018
edited by CPhill  Dec 25, 2018

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