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Based on The Erasmus polyhedron by @Gamow..

Construct a 3D convex polyhedron of any single material that can float in water such that $x\%$ of its volume is below water level and $y\%$ of its surface area is above water level. Find the solution with maximum $(x\times y)$.

Notes:

  • Archimedes principle is valid.

  • Material must be something that exists in real life.

  • Assume that it does not have any holes/pores.

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    $\begingroup$ Note that if anyone's answer is more than 10000, we are done. $\endgroup$
    – the4seasons
    Sep 27 '15 at 9:08
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Solution with $xy \to 5000$.

To maximise the volume in the water, choose a material that has the same density as water, and place it so that the top face sits at the surface of the water. Having equal density with water means the object won't rise or sink. This allows $x$ to approach 100.

To maximise the surface area above the water, choose a short shape with one large face. An extremely shallow pyramid would allow $y$ to approach 50. (I'm ignoring the effects of the water's surface tension.)

Water density varies with temperature, with a peak of 1000 kgm$^{-3}$ at 4$^\text{o}$C. At 80$^\text{o}$C, the density of water is 971.8 kgm$^{-3}$, which is close to sodium's 970 kgm$^{-3}$. (Don't actually put a slab of sodium into hot water, though!)

Plastics come in a variety of densities, with Polyethylene Polystone M-Flametech said to have a density of 0.98 g/cm$^3$, or 980 kgm$^{-3}$. A slab of this material should do the trick with the water somewhere between 60$^\text{o}$C and 80$^\text{o}$C.

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    $\begingroup$ But would it float stably? $\endgroup$
    – lirtosiast
    Sep 28 '15 at 5:02
  • $\begingroup$ And are you convinced that this is the best? Why choose a pyramid, not a cone? $\endgroup$
    – ghosts_in_the_code
    Sep 28 '15 at 7:52
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    $\begingroup$ @ghosts_in_the_code I'm not sure. It's certainly the greatest $y$ for $x \to 100$. With just one face facing up and exposed, the other faces (all submerged) must add up to at least the same surface area. However, if $x$ was smaller, it might be possible for $y$ to be disproportionately larger to increase $xy$. On the second question, you asked for a polyhedron, which I take to mean a 3D shape whose faces are all flat. A cone fails that definition, though it could be considered the limit of a sequence of pyramids with an increasing number of sides. $\endgroup$
    – Lawrence
    Sep 28 '15 at 8:48

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