Out of a circular piece of paper, you wish to form a cone cup, so you cut out a circle wedge (with its extreme at the circle center) and join the resulting straight sides, forming a conical cup.

What size should the wedge be, to maximize the capacity of the cone?

"Another Answer" below gives the answer (105.5 degrees) that maximizes the area of the central triangle of the cone.  Notice how much different it is than the answer (66.06 degrees) that maximizes the volume of the cone.  One's first thought is that these two answers ought to be the same, or at least a lot closer than they turn out to be. But I think each is correct for what it maximizes, so intuition is foiled again.

Posted by Richard on 2005-04-02 04:04:32