My National Review Online
"Diary" column for January 2006 included the
following "senseless problem."
Senseless Problem
——————
There is a ball 12
feet in diameter on top of a pole 60 feet high. On the ball stands a man
whose eye is 6 ft above the ball. How much ground beneath the ball is
invisible to him?
———————————————
Too easy. The man's eye is at the apex of a triangle 78 feet high
(60 + 12 + 6). A tangent to the sphere from this point makes an
angle of 30 degrees with the vertical. That makes the triangle
equilateral—all sides equal, all angles 60 degrees, height to base in the
ratio
Ö3
to 2. Half the triangle's base is therefore 78 /
Ö3,
which is equal to 26Ö3.
The ground invisible to the man is a circle with this radius, area 2028p.
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