## 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 `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`.

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