Problem of the Week #6: Monday Feb. 13th, 2023 As before, these problems are the results of me following my curiosity, and I make no promises regarding the topics, difficulty, solvability of these problems. Please register for an account if you would like to join the discussion below or share your own problems.
a). If we were to cut a cone, of base radius r and height h, parallel to its base so that the top portion and the bottom portion have the same volume, how far from the top (pointy end) should we cut?
b). If we were to cut a frustum of a cone (a cone with its top cut off parallel to its base), with bottom radius R, top radius r, and height h, parallel to its base so that the top and bottom portions have the same volume, how far from the top (smaller circle) should we cut?
c). If we were to cut a hemisphere, of radius r, parallel to its base so that the top and bottom portions have the same volume, how far from the top (spherical end) should we cut?
d). If we were to cut a frustum of a hemisphere (a hemisphere with its top cut off parallel to its base), with bottom radius R and top radius r, parallel to its base so that the top and bottom portions have the same volume, how far from the top (smaller circle) should we cut?
e). Share your own problem inspired by these considerations.
f). Show one of these questions to a friend/colleague/student/family member to start a mathematical discussion.