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?

Call Charlie's ratio from one marble's radius to the next aa=sqrt(3/2). Let r1 be the radius of the bottom (first) marble and r2 be the radius of the next (second) marble up. Then the distance from the two centers is r1+r2 and r2/r1=aa.
Draw in a radial line segments from the marbles' centers to respective tangent points formed with the funnel. Notice that both of these segments form right angles with the funnel. Now draw a line segment from the first marble's center perpendicular to the second marble's radial segment you just drew. We now have a right triangle with the centers at the two non-right vertices. The hypotenuse is of length r1+r2 and vertical with respect to the funnel wall, one leg is of length r1-r2 and the last leg is parallel to the funnel wall. Thus the angle at the first marble's center is the same angle that is formed by the funnel wall from vertical.
This angle is arcsin((r2-r1)/(r2+r1))=arcsin((aa-1)/(aa+1))=arcsin(5-2sqrt(6)); almost 6 degrees. Mighty steep.

Posted by owl on 2005-06-29 21:48:14