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January 30, 2015

Posted by **Paul** on Sunday, April 24, 2011 at 6:39pm.

Thanks,

Paul

- Science/Math -
**Damon**, Sunday, April 24, 2011 at 6:58pmhemisphere boat has max volume (buoyancy by Archimedes principle) for given surface area.

sphere V = (4/3) pi r^3

area A = 4 pi r^2

V/A = r/3 so you make the radius as big as possible, spreading the clay out so it almost leaks

To show that a sphere is optimum is more complicated. The easy way is to do a thought experiment. Compare the surface area of a sphere to the area of a cube with the same volume (6 x^2). The next way is to perturb the sphere with higher harmonics of Bessel functions and show that the area gets bigger when the shape is perturbed.

- Science/Math -
**MathMate**, Sunday, April 24, 2011 at 6:59pmBy Archimedes principle, the mass of water displaced equals the mass of a floating object.

Therefore to have the maximum mass of the boat (plus its load) that remains floating means to maximum volume of the boat.

For the displacement of a boat, the minimum submerged surface area is a hemisphere. This means that if you make the boat a hemisphere, with as large radius as you can make it without breaking or overloading the clay shell, the boat will carry the heaviest cargo.

- Science/Math -
**Damon**, Sunday, April 24, 2011 at 7:02pm - Science/Math -
**Damon**, Sunday, April 24, 2011 at 7:10pmI can not find a proof immediately online. The link below gives a rationale:

http://en.wikipedia.org/wiki/Sphere

excerpt:

7. Of all the solids having a given volume, the sphere is the one with the smallest surface area; of all solids having a given surface area, the sphere is the one having the greatest volume.

These properties define the sphere uniquely. These properties can be seen by observing soap bubbles. A soap bubble will enclose a fixed volume and due to surface tension its surface area is minimal for that volume. This is why a free floating soap bubble approximates a sphere (though external forces such as gravity will distort the bubble's shape slightly).

- Science/Math -
**Damon**, Sunday, April 24, 2011 at 8:59pmHere are some more attempted proofs:

http://answers.yahoo.com/question/index?qid=20071019213552AANdb52

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