# Science/Math

posted by .

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

• Science/Math -

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.

• Science/Math -

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.

• Science/Math -

http://en.wikipedia.org/wiki/Surface-area-to-volume_ratio

• Science/Math -

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).

• Science/Math -

Here are some more attempted proofs:

## Similar Questions

1. ### Math! Help! Help! Help!

Hello a compound with: density of: .80 g/cm^(3) volume of: 1ML what is the mass?
2. ### Science

Felipe made a toy boat out of clay, but it kept sinking in water. He changed the boat's design by making it slightly wider than before. Why did Felipe's newly designed boat float on water?
3. ### Science ~CHECK MY WORK PLEASE~

Felipe made a toy boat out of clay, but it kept sinking in water. He changed the boat's design by making it slightly wider than before. Why did Felipe's newly designed boat float on water?
4. ### science ~CHECK MY WORK PLEASE~

1) Both carbon dioxide and water are pure substances. How are they similar to each other?
5. ### Science ~ Structures and Properties of Matter

1) Both carbon dioxide and water are pure substances. How are they similar to each other?
6. ### Science

Felipe made a toy boat out of clay, but it kept sinking in water. He changed the boat's design by making it slightly wider than before. Why did Felipe's newly designed boat float on water?
7. ### Science

Felipe made a toy boat out of clay, but it kept sinking in water. He changed the boat's design by making it slightly wider than before. Why did Felipe's newly designed boat float on water?
8. ### Science

Felipe made a toy boat out of clay, but it kept sinking in water. He changed the boat's design by making it slightly wider than before. Why did Felipe's newly designed boat float on water?
9. ### science

Felipe made a toy boat out of clay, but it kept sinking in water. He changed the boat's design by making it slightly wider than before. Why did Felipe's newly designed boat float on water?
10. ### Physics

In an immersion measurement of a boy’s density, he is found to have an apparent weight of 6.86-N when completely submerged in water with lungs empty. The mass of the boy is 68.0-kg in air. The density of water is 1000-kg/m^3. a)Determine …

More Similar Questions