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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 r) = 32/8 = 4.

12 + r = 48 4r

5r = 36 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.