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.
Kenny M forgot about the third type of circle: the slanted type.
A torus is defined by major radius R and minor radius r.
There are 3 types of circles.
A. The ones parallel to the axis which all have radius r
B. The ones perpendicular to the axis and can have any radius from R-r to R+r.
C. The slanted Villarceau circles which have radius R.
https://mathworld.wolfram.com/VillarceauCircles.html
Any circle of one type crosses every circle of the other two types.
Jack and Hannah therefore have type B.
Ron has type C. R=60/(2pi)
Dana has type A. r=50/(2pi)
The largest radius is then a type B with radius R+r=110/(2pi)
and circumference 110.
This desmos3D came out pretty nice.
https://www.desmos.com/3d/lz4zlzlk7o
I scaled it down by a factor of 20pi so R=6, r=5. Feel free to play with the sliders.
I arbitrarily placed Jack's circle at the top.
Edited on December 3, 2024, 3:23 pm
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Posted by Jer
on 2024-12-03 15:12:35 |
Solution
