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Many Marbles Mystery (Posted on 2005-06-29) Difficulty: 3 of 5
Five marbles of descending sizes are placed in a conical funnel. Each marble is in contact with the adjacent marble(s). Also, each marble is in contact all around the funnel wall.

If the smallest marble has a radius of 8mm, and the largest marble has a radius of 18mm, what is the radius of the middle marble?

Bonus question, suggested by "Juggler": what's the angle of the funnel walls?

See The Solution Submitted by Old Original Oskar!    
Rating: 4.0000 (2 votes)

Comments: ( Back to comment list | You must be logged in to post comments.)
Solution Solution | Comment 3 of 4 |
 
Let r5 > r4 > r3 > r2 > r1 > 0 be the 
radii of the marbles. From similar triangles
we get the following equations:
  r2 - r1     r3 - r2     r4 - r3     r5 - r4
 --------- = --------- = --------- = ---------
  r2 + r1     r3 + r2     r4 + r3     r5 + r4
Simplifying we get the following:
  r2*r2 = r1*r3
  r3*r3 = r2*r4
  r4*r4 = r3*r5
Therefore,
  (r3*r3)*(r3*r3) = (r2*r4)*(r2*r4)
                
                  = (r2*r2)*(r4*r4)
                  = (r1*r3)*(r3*r5)
Hence,
  r3 = sqrt(r1*r5)
For our problem,
  r3 = sqrt(8*18) = sqrt(144) = 12 mm.
For the angle alpha (between the funnel
wall and the axis of the funnel),
                r2 - r1     sqrt(r1*r3) - r1 
  sin(alpha) = --------- = ------------------
                r2 + r1     sqrt(r1*r3) + r1
                sqrt(8*12) - 8
             = ----------------
                sqrt(8*12) + 8
             ~= 0.1010205  
           or
  alpha ~= 5.7979393 degrees 
 

  Posted by Bractals on 2005-06-30 05:03:36
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