This article is from the Puzzles FAQ, by Chris Cole firstname.lastname@example.org and Matthew Daly email@example.com with numerous contributions by others.
You are hiking in a half-planar woods, exactly 1 mile from the edge,
when you suddenly trip and lose your sense of direction. What's the
shortest path that's guaranteed to take you out of the woods? Assume
that you can navigate perfectly relative to your current location and
Go 2/sqrt(3) away from the starting point, turn 120 degrees and head
1/sqrt(3) along a tangent to the unit circle, then traverse an arc of
length 7*pi/6 along this circle, then head off on a tangent 1 mile.
This gives a minimum of sqrt(3) + 7*pi/6 + 1 = 6.397...
It remains to prove this is the optimal answer.
Old Boniface he took his cheer,
Then he bored a hole through a solid sphere,
Clear through the center, straight and strong,
And the hole was just six inches long.
Now tell me, when the end was gained,
What volume in the sphere remained?
Sounds like I haven't told enough,
But I have, and the answer isn't tough!
The volume of the leftover material is equal to the volume of a 6" sphere.
First, lets look at the 2 dimensional equivalent of this problem. Two
concentric circles where the chord of the outer circle that is tangent
to the inner circle has length D. What is the annular area between the
It is pi * (D/2)^2. The same area as a circle with that diameter.
big circle radius is R little circle radius is r 2 2 area of donut = pi * R - pi * r 2 2 = pi * (R - r ) Draw a right triangle and apply the Pythagorean Theorem to see that 2 2 2 R - r = (D/2) so the area is 2 = pi * (D/2)
radius = srqt(3^2 - h^2) so the area is pi * ( 3^2 - h^2 )
pi * ( R^2 - 3^2 ) the area of the doughnut is therefore pi(R^2 - h^2) - pi( R^2 - 3^2 ) = pi (R^2 - h^2 - R^2 + 3^2) = pi (3^2 - h^2)