## 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?

## 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).

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

## 62 - 63 Maxima and minima: cylinder inscribed in a cone and cone inscribed in a sphere

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.

## 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.

## 34 Review Problem - Sphere dropped into a cone

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?

## 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. ## Solids for which V = (1/3)Bh

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. 