cone

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.
 

Problem 13
A sphere is cut in the shape of a circular cone. How much of the material can be saved?
 

Problem 12
Find the altitude of the circular cone of maximum convex surface inscribed a sphere of radius a.
 

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?
 

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

Problem 65
Find the circular cone of minimum volume circumscribed about a sphere of radius a.
 

Problem 62
Inscribe a circular cylinder of maximum convex surface area in a given circular cone.
 

Problem 63
Find the circular cone of maximum volume inscribed in a sphere of radius a.

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.
 

Problem 34
The inside of a vase is an inverted cone 2.983 in. across the top and 5.016 in. deep. If a heavy sphere 2.498 in. in diameter is dropped into it when the vase is full of water, how much water will overflow?
 

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

This is a group of solids in which the volume is equal to one-third of the product of base area and altitude. There are two solids that belong to this group; the pyramid and the cone.