Sometime in the future, five private
astronauts are exploring a
recently discovered solar system
and land on one of its small planets. To their amazement, they discover the planet is in the shape of
a perfect torus.
Furthermore, they detect a series of channels, each of which is a perfect circle formed by the intersection of a plane with the planet’s surface.
Suspecting that the channels were dug by
intelligent beings, the five start out
looking for artifacts.
Hannah follows one channel and finds its length to be 30 kms in circumference.
Jack follows a longer channel which
does not cross Hannah’s.
Dana’s channel is 50 kms and crosses Jack’s.
Ron’s channel is 60 kms and also crosses Jack’s.
Finally, Sarah follows a channel
which is the maximum possible
circumference.
Determine the length of Sarah’s
channel.
Let r be the radius of the circular section that is rotated to create the torus, and R be radius used to rotate the circle section.
If the plane is parallel to and includes the axis of rotation of the torus, then one can create an infinite number of circular channels of negligible width just by rotating the plane around the torus. These would all be radius r.
Next, the word "channel" I believe is significant.
The only other way, I believe, to create additional circular "channels' Is to make the plane perpendicular to the axis of rotation. Then one can create circular "channels" of mean radius R and width 2*e where 0<=e<=r.
Maybe more later after I think about this more.
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Posted by Kenny M
on 2024-12-03 07:08:08 |
Some thoughts - not a solution
