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.
 

Problem 67
An Indian tepee is made by stretching skins or birch bark over a group of poles tied together at the top. If poles of given length are to be used, what shape gives maximum volume?
 

58 - 59 Maxima and minima: Cylinder surmounted by hemisphere and cone

Problem 58
For the silo of Problem 57, find the most economical proportions, if the floor is twice as expensive as the walls, per unit area, and the roof is three times as expensive as the walls, per unit area.
 

Problem 59
An oil can consists of a cylinder surmounted by a cone. If the diameter of the cone is five-sixths of its height, find the most economical proportions.
 

The Cone

Cone
The surface generated by a moving straight line (generator) which always passes through a fixed point (vertex) and always intersects a fixed plane curve (directrix) is called conical surface. Cone is a solid bounded by a conical surface whose directrix is a closed curve, and a plane which cuts all the elements. The conical surface is the lateral area of the cone and the plane which cuts all the elements is the base of the cone.

 

Like pyramids, cones are generally classified according to their bases.
 

Right cone and Oblique Cone with Some Elements