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Bag Of Balls

Problem

Three balls are placed inside a cone such that each ball is in contact with the edge of the cone and the next ball.


If the radii of the balls are 20 cm, 12 cm, and r cm respectively, what is the value of r?


Solution

Consider the following diagram.


Using similar triangles, (12 + r)/(12 minus r) = 32/8 = 4.

therefore 12 + r = 48 minus 4r

therefore 5r = 36 implies r = 7.2 cm.

What if the radii of the three balls were 20 cm, r cm, and 10 cm?
If the top sphere has radius, x, and the bottom sphere has radius, y, find r in terms of x and y.

Problem ID: 139 (Dec 2003)     Difficulty: 3 Star

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