## 23 - Sphere cut into a pyramid

**Problem 23**

A sphere is cut in the form of a right pyramid with a square base. How much of the material can be saved?

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**Problem 23**

A sphere is cut in the form of a right pyramid with a square base. How much of the material can be saved?

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**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.

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**Problem 3**

Find the weight of the heaviest circular cylinder can be cut from a 16-lb shot.

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**Problem 02**

A cylinder is inscribed in a given sphere. Find the shape of the cylinder if its convex surface area is a maximum.

**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 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.

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**Problem 36**

A hole 6 in. in diameter was bored through a sphere 10 in. in diameter. Find the volume of the part cut out.

**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?

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