THE BALL PROBLEM.
(
Patchwork Puzzles)
A stonemason was engaged the other day in cutting out a round ball for
the purpose of some architectural decoration, when a smart schoolboy
came upon the scene.
"Look here," said the mason, "you seem to be a sharp youngster, can you
tell me this? If I placed this ball on the level ground, how many other
balls of the same size could I lay around it (also on the ground) so
that every ball should touch this one?"
The boy at once gave the correct answer, and then put this little
question to the mason:--
"If the surface of that ball contained just as many square feet as its
volume contained cubic feet, what would be the length of its diameter?"
The stonemason could not give an answer. Could you have replied
correctly to the mason's and the boy's questions?
Answer:
If a round ball is placed on the level ground, six similar balls may be
placed round it (all on the ground), so that they shall all touch the
central ball.
As for the second question, the ratio of the diameter of a circle to its
circumference we call _pi_; and though we cannot express this ratio in
exact numbers, we can get sufficiently near to it for all practical
purposes. However, in this case it is not necessary to know the value of
_pi_ at all. Because, to find the area of the surface of a sphere we
multiply the square of the diameter by _pi_; to find the volume of a
sphere we multiply the cube of the diameter by one-sixth of _pi_.
Therefore we may ignore _pi_, and have merely to seek a number whose
square shall equal one-sixth of its cube. This number is obviously 6.
Therefore the ball was 6 ft. in diameter, for the area of its surface
will be 36 times _pi_ in square feet, and its volume also 36 times _pi_
in cubic feet.
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