What is a Conical Frustum?

Geometry Level 3

Imagine an perfect hourglass that has a base radius \(R_1\) and a height \(Z\) from base to center. A solid ball with a radius of \(R_2\) is stuck perfectly within the middle of the hourglass (it leaves no gaps). The hourglass is filled completely with sand on both sides of the hourglass. Given that

  • \(Z = \dfrac {3}{R_1R_2\pi} \text{ cm}\).
  • \(R_1+R_2 = 9\sqrt {R_1R_2}\text{ cm}\).
  • The volume of the sand is \(120\text{ cm}^3\).

Find the volume of the ball.

Note: Image drawn not up to scale.


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