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?
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 20050630 05:03:36 