Cylinders

Cylinders have more surface area per unit of volume than spheres. For convenience, (and because it's a common ratio in commercial packing materials), we'll assume that our cylinders are twice as long as they're wide, so if the diameter is d then the length is 2d.

Also, to make the argument easier to follow, we'll make each cylinder out of one of the marbles. Doing the calculations, the width of each cylinder is 1/31/3 times the diameter of the marble, or d = D / 31/3 = 0.69336D. More math shows that the surface of our cylinder has an area of

2.5TCd2 = 3.776cm2, an improvement over the marble which had a surface area of K = 3.1416 cm . The volume of each cylinder is ftd3 / 2, and assuming the same RCP factor of 0.6 (it is actually better with cylinders, but we're being conservative), NTCd3 / 2 = 0.6 V, so NTCd2 / 0.6 V = 2 / d Thus the ratio Z = 2.5NTCd2 / 0.6V = 5 / d = 5* 31/3 / D = 7.2 / D (the marble was 3.6 / D)

The ratio Z for the cylinder is twice that of the marble with the same volume!

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