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Spheres on a floor (Posted on 2019-02-07) Difficulty: 5 of 5
There are 4 solid spheres arranged so that each one is touching all of the others. The 3 bottom spheres touch the flat floor at points A, B and C. The top sphere has a radius of 12 centimeters. If it were replaced by a sphere with radius 25 cm, then its center would be 14, 15 and 16 cm further from from points A, B and C, respectively.

What is the radius of each sphere?

No Solution Yet Submitted by Danish Ahmed Khan    
Rating: 5.0000 (1 votes)

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Since this is D5... | Comment 1 of 2
I decided to share what I've tackled so far.

If two spheres of radius a and b are sitting on a floor touching each other, the distance from the points they touch the floor is AB=2sqrt(ab).  

For 3 spheres, their floor points then form a triangle ABC with 3 known side lengths.  
Then angles of this triangle can be found with the law of cosines:  cos(BAC)=(ac+ab-bc)/(2a sqrt(bc)).

These three spheres can then be given equations.  Here's a top view with centers at (0,0,a) (2sqrt(ab),0,b) (messy,messier,c):

I don't know how to get the 4th sphere to sit on top.

Once I figure this out it will be a lot more algebra.

  Posted by Jer on 2019-02-07 21:30:14
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