Science/Math
posted by Paul .
We have 30 grams of clay. We understand that Density=Mass/Volume. We understand that the Density of water is 1.0 and in order for the clay boat to float in the water Density needs to be less than 1.0. So my question is using 30 grams of clay what design will allow me to hold the most pennies and still have the clay boat float? We understand that we are trying to build the shape that gives us the most volume, but we are trying to mathematically come up with the perfect dimensions. Please help.
Thanks,
Paul

hemisphere 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. 
By 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. 

I 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). 
Here are some more attempted proofs:
http://answers.yahoo.com/question/index?qid=20071019213552AANdb52