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 Many Marbles Mystery (Posted on 2005-06-29)
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)

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 Solution | Comment 3 of 4 |
` `
`Let r5 > r4 > r3 > r2 > r1 > 0 be the radii of the marbles. From similar triangleswe 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 funnelwall 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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