22 - Smallest cone that may circumscribe a sphere

Problem 22
A sphere of radius a is dropped into a conical vessel full of water. Find the altitude of the smallest cone that will permit the sphere to be entirely submerged.

03 - Heaviest cylinder that can be made from a shot

Problem 3
Find the weight of the heaviest circular cylinder can be cut from a 16-lb shot.

02 - Cylinder of maximum convex area inscribed in a sphere

Problem 02
A cylinder is inscribed in a given sphere. Find the shape of the cylinder if its convex surface area is a maximum.

66 - 68 Maxima and minima: Pyramid inscribed in a sphere and Indian tepee

Problem 66
Find the largest right pyramid with a square base that can be inscribed in a sphere of radius a.

64 - 65 Maxima and minima: cone inscribed in a sphere and cone circumscribed about a sphere

Problem 64
A sphere is cut to the shape of a circular cone. How much of the material can be saved? (See Problem 63).