Eight Little

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Eight Little
Eight Little
Eight identical little spheres of mercury coalesce to form a single sphere. Compared with the combined...?


Eight identical little spheres of mercury coalesce to form a single sphere. Compared with the combined surface areas of the eight little spheres, the surface area of the big sphere is...

A. one eight.
B. one quarter.
C. one half.
D. the same.
E. double.

Help me!

The large sphere will have double the radius of one of the 8 individual small spheres.

Volume = 4/3 pi r^3
(see that if you double the radius, the volume goes up x8)

That means that the one large sphere will have 4 times the surface area of one of the smaller spheres.

Surface area = 4pi r^2
(see that if you double the radius in this case, the surface area goes up x4)

For equal volumes:
1 large sphere (4 square units of area) (total: 4)
8 small spheres (1 square unit of area each) (total: 8)

So in the overall picture, the one larger sphere will have half of the surface area of the 8 smaller spheres, for the same volume.

Answer:
C. one half



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2 comments to Eight Little

  • Cal King

    In Eight Little Piggies, Gould tries to usurp the usual assumptions about evolution: that it will lead ultimately to perfection. If, for example, all tetrapods have an ancestor with five digits on their hands and feet, then five digits must represent perfection, a product of natural selection and the struggle for survival. Gould argues that since the earliest known tetrapods had more than five, we could have ended up with more or fewer digits than five as the norm for all tetrapods, and that there is nothing special about five digits. It may simply be an accident of past history.

    He did not elaborate on the scenario on how five digits might have arisen as the de facto norm, but I can suggest one possible scenario. Gould did suggest that early tetrapods may have experimented with different digital counts and then "stabilized" at five. Let's assume that the first tetrapod had several descendant lineages, and each descendant lineage may have the same or slightly different numbers of digits. Since these lineages are all similar in being tetrapods, they may have competed with each other, for food for instance. For some reason other than the number of digits, one species was the winner and survived while other lineages withered away, and if this species happened to have 5 digits, then all of its descendants will have five digits. The lineages having a different digital number vanished, not because they have a different number of digits than 5, but because they were not successful in the struggle of life or they were not lucky enough to have the right set of adaptations when the environment changed.

    In sum, not all of our body parts are honed to perfection by evolution, which is what some biologists and laymen alike may assume. Some of the features we have are simply the products of chance, or contingency, the byproducts of the struggle for survival. Go back in time, and rerun history, and we may end up with a different outcome and a different number of digits for tetrapods. Or we may not have any tetrapods at all.

    Gould also presented evidence to debunk the idea that five is the ideal number of digits, since many organisms (such as the panda) have extra "digits," making a total of six on the their hand or feet. But of course the panda's thumb is not a true digit since it is not formed by adding an additional digit next to the thumb during development but by elongating another bone that is not normally involved in forming digits. Of course there are also many animals with fewer than 5 digits. Hence in doing so, he lent further support that pentadactyly is not an optimal number that evolved due to natural selction, which would be the default position of the panselectionists, but that there is room for chance to play a role in evolutionary history.