__Category__: Solid Geometry"Published in Newark, California, USA"

An ink bottle is in the form of a right circular cylinder with a large conical opening as shown in the figure. When it is filled with the bottom of the opening, it can just be turned upside down without any ink spilling. Prove that the depth of the cone is

^{3}/

_{5}the depth of the bottle.

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__Solution__:

When you flipped the ink bottle, there should be no spills of ink as shown below

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The vertical section of the above figure will be like this

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As you can see in the figure that the level of the ink is exactly at the tip of the right circular cone. If it is above the tip of the right circular cone, then there will be the spillage of the ink. Let's label further the given figure as shown below

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Volume before Inversion = Volume after Inversion

but

The above equation becomes

Therefore