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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 4$r$

5$r$ = 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 $y$.