Some years ago I taught a calculus course for especially talented students. I made up the following problem as an interesting challenge.
Draw a unit disk on paper. Cut out a sector of some angle α from the disk. Roll each of the resulting two pieces into cones. Let V(α) be the sum of the volumes of the two cones. Find the α that maximizes V(α).
Here is an animated statement of the problem, produced in Maple.
