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?
 

29 - 31 Solved problems in maxima and minima

Problem 29
The sum of the length and girth of a container of square cross section is a inches. Find the maximum volume.
 

Problem 30
Find the proportion of the circular cylinder of largest volume that can be inscribed in a given sphere.
 

Problem 31
In Problem 30 above, find the shape of the circular cylinder if its convex surface area is to be a maximum.