Transcription
A paper cone is filled with water to exactly half of its total height. What fraction of the cone's total volume is actually filled with water, though? You have 5 seconds to answer. 5, 4, 3, 2, 1.
If you guessed 1/2, you fell for the trap. Let's look at the actual geometry of the cone. As you move halfway down the cone, the height is cut in half, but the circular radius is also cut in half. Volume [clears throat] in three dimensions relies on the square-cube law.
Take a look at the actual volume formula for a cone. It's 1/3 pi times the radius squared times the height. So, if the height drops by a factor of 1/2 and the radius drops by a factor of 1/2, watch what happens to the math. The new volume is now 1/3 pi times 1/2 the radius squared times 1/2 the weight. When you square that 1/2 radius, it becomes 1/4. And 1/4 multiplied by that 1/2 height becomes 1/8.
The small water chunk at the bottom does not hold half the volume. It holds exactly 1/8 of the total volume. If we freeze that bottom chunk of water and pull it out of the cone, we can duplicate it. Watch as we physically stack them together. 1, 2, 3, 4, 5, 6, 7, 8. It takes exactly eight of those bottom chunks to perfectly fill the original.