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.

Solution:

 

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?

Solution:

 

Problem 68
Solve Problem 67 above if poles of any length can be found, but only limited amount of covering material is available.

Solution:

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