EXERCISE 98

MISCELLANEOUS PROBLEMS

1. The slant height of the frustum of a regular pyramid is 25 ft., and the sides of its square bases are 54 ft. and 24 ft. respectively. Find the volume.

2. If the bases of the frustum of a pyramid are regular hexagons whose sides are 1 ft. and 2 ft. respectively, and the volume of the frustum is 12 cu. ft., find the altitude.

3. From a right circular cone whose slant height is 30 ft., and the circumference of whose base is 10 ft., there is cut off by a plane parallel to the base a cone whose slant height is 6 ft. Find the lateral area and the volume of the frustum.

4. Find the difference between the volume of the frustum of a pyramid whose altitude is 9 ft. and whose bases are squares, 8 ft. and 6 ft. respectively on a side, and the volume of a prism of the same altitude whose base is a section of the frustum parallel to its bases and equidistant from them.

5. A Dutch stone windmill in the shape of the frustum of a right cone is 12 meters high. The outer diameters at the bottom and the top are 16 meters and 12 meters, the inner diameters 12 meters and 10 meters. How many cubic meters of stone were required to build it?

6. The chimney of a factory has the shape of a frustum of a regular pyramid. Its height is 180 ft., and its upper and lower bases are squares whose sides are 10 ft. and 16 ft. respectively. The flue throughout is a square whose side is 7 ft. How many cubic feet of material does the chimney contain?

7. Two right triangles with bases 15 in. and 21 in., and with hypotenuses 25 in. and 35 in. respectively, revolve about their third sides. Find the ratio of the total areas of the solids generated and find their volumes.

EXERCISE 99

EQUIVALENT SOLIDS

1. A cube each edge of which is 12 in. is transformed into a right prism whose base is a rectangle 16 in. long and 12 in. wide. Find the height of the prism and the difference between its total area and the total area of the cube.

2. The dimensions of a rectangular parallelepiped are a, b, c. Find the height of an equivalent right circular cylinder, having a for the radius of its base; the height of an equivalent right circular cone having a for the radius of its base.

3. A regular pyramid 12 ft. high is transformed into a regular prism with an equivalent base. Find the height of the prism.

4. The diameter of a cylinder is 14 ft. and its height 8 ft. Find the height of an equivalent right prism, the base of which is a square with a side 4 ft. long.

5. If one edge of a cube is e, what is the height h of an equivalent right circular cylinder whose radius is r?

6. The heights of two equivalent right circular cylinders are in the ratio 4:9. If the diameter of the first is 6 ft., what is the diameter of the second?

7. A right circular cylinder 6 ft. in diameter is equivalent to a right circular cone 7 ft. in diameter. If the height of the cone is 8 ft., what is the height of the cylinder?

8. The frustum of a regular pyramid 6 ft. high has for bases squares 5 ft. and 8 ft. on a side. Find the height of an equivalent regular pyramid whose base is a square 12 ft. on a side.

9. The frustum of a cone of revolution is 5 ft. high and the diameters of its bases are 2 ft. and 3 ft. respectively. Find the height of an equivalent right circular cylinder whose base is equal in area to the section of the frustum made by a planę parallel to the bases and equidistant from them.

EXERCISE 100

REVIEW QUESTIONS

1. Define polyhedron. Is a cylinder a polyhedron ?

2. Define prism, and classify prisms according to their bases. 3. How is the lateral area of a prism computed? Is the method the same for right as for oblique prisms?

4. Define parallelepiped; rectangular parallelepiped; cube. Is a rectangular parallelepiped always a cube? Is a cube always a rectangular parallelepiped?

5. Distinguish between equivalent and congruent solids. Are two cubes with the same altitudes always equivalent? always congruent? Is this true for parallelepipeds?

6. What are the conditions of congruence of two prisms? of two right prisms? of two cubes?

7. The opposite angles of a parallelogram are equal. What is a corresponding proposition concerning parallelepipeds?

8. How do you find the volume of a parallelepiped? What is the corresponding proposition in plane geometry?

9. How do you find the volume of a prism? of a cylinder? of a pyramid? of a cone?

10. Define pyramid. How many bases has a pyramid? Is there any kind of a pyramid in which more than one face may be taken as the base?

11. How do you find the lateral area of a pyramid? of a right cone? of a frustum of a pyramid? of a frustum of a right cone?

12. How many regular convex polyhedrons are possible? What are their names?

13. Given the radius of the base and the altitude of a cone of revolution, how do you find the volume? the lateral area? the total area?

BOOK VIII

THE SPHERE

619. Sphere. A solid bounded by a surface all points of which are equidistant from a point within is called a sphere.

The point within, from which all points on the surface are equally distant, is called the center. The surface is called the spherical surface, and sometimes the sphere. Half of a sphere is called a hemisphere. The terms radius and diameter are used as in the case of a circle.

620. Generation of a Spherical Surface. By the definition of sphere it appears that a spherical surface may be generated by the revolution of a semicircle about its diameter as an axis.

Thus, if the semicircle ACB revolves about AB, a spherical surface is generated. It is therefore assumed that a sphere may be described with any given point as a center and any given line as a radius.

621. Equality of Radii and Diameters. It follows that:

All radii of the same sphere are equal, and all diameters of the same sphere are equal.

Equal spheres have equal radii, and spheres having equal radii are equal.

PROPOSITION I. THEOREM

622. Every intersection of a spherical surface by a plane is a circle.

Given a sphere with center O, and ABD any section of its surface made by a plane.

To prove that the section ABD is a circle.

Proof. Draw the radii OA, OB, to any two points A, B, in the section, and draw OC L'to the plane of the section. Then in AOCA and OCB, OCA and OCB are rt. 4,

OC is common, and OA = OB.

.. AOCA is congruent to AOCB.

.. CA CB.

§ 430

§ 621

$ 89

§ 67

.. any points A and B, and hence all points, in the section are equidistant from C, and ABD is a O, by $159.

Q. E.D.

623. COROLLARY 1. The line joining the center of a sphere and the center of a circle of the sphere is perpendicular to the plane of the circle.

624. COROLLARY 2. Circles of a sphere made by planes equidistant from the center are equal; and of two circles made by planes not equidistant from the center the one made by the plane nearer the center is the greater.