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Fraction of a Can

Fill a cylindrical can with water. A quick way to empty the can to exactly
half its volume is to pour out the water until the water line just touches the
circular bottom of the can. If you tip the can further, until the water
line crosses the center of the bottom of the can (forming a diameter of the
bottom circle), what fraction of the can's volume is now filled with water?
Source: Original.

Solutions were receive from Radu Ionescu, Dan Dima, Richard Mathar, Dan
Chirica, Nick McGrath, David Bachtel, Kirk Bresniker, Sean Forbes, Alan
O'Donnell, Jeremy Galvagni, Yucel Ozcan, Ramneek Sharma, Philippe
Fondanaiche, Raśl Mora Pastor. Many solvers sent the (incorrect)
solution of half the volume of a cone: 1/6. Most then attacked the
problem again and found the needed integrals to solve the problem. I
didn't realize that an application of calculus would be required. Kirk
Bresniker found a reference for this problem at
http://mathworld.wolfram.com/CylindricalWedge.html. The solution
below is from Raśl Mora Pastor.

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