8. Allowing 490 lb. per cubic foot, find to the nearest 0.01 lb. the weight of a steel plate 5 ft. by 4 ft. 6 in. by 1ğ in.

9. Through a steel plate 5 ft. long, 3 ft. 8 in. wide, and in. thick, a porthole 12 in. in diameter is cut. Allowing 0.29 lb. per cubic inch, find the weight of the finished plate.

10. A cast-iron base for a column is in the form of a frustum of a regular pyramid. The lower base is a square 26 in. on a side, the upper base has one fourth the area of the lower base, and the altitude of the frustum is 10 in. If the total surface is to be painted, what area must be covered? 11. A cylinder head for a steam engine has the shape shown in this figure. Allowing 41 lb. for the weight of a steel plate 1 ft. square and 1 in. thick, find the weight of the cylinder head to the nearest 0.1 lb.

127

12. A hollow steel shaft 14 ft. long has an exterior diameter of 16 in. and an interior diameter of 9 in. Allowing 0.29 lb. per cubic inch, find the weight to the nearest pound. 13. A steel beam in the form here shown is 16 ft. long. Allowing 0.29 lb. per cubic inch, find the weight of the beam to the nearest pound.

14. How many square feet of tin are required to make a funnel, if the diameters at the top and bottom are to be 32 in. and 16 in. respectively, the height is to be 24 in., and 4 sq. in. are allowed for overlapping?

15. Find the cost, at $1.35 per square foot, of finishing the curve surface of a frustum of a right circular cone whose slant height is 12 ft. and the radii of whose bases are 3 ft. 6 in. and 2 ft. 4 in. respectively.

Exercises. Equivalent Solids

1. When a cube of iron 6 in. on an edge is melted it just fills a mold in the form of a right prism whose base is a rectangle 8 in. long and 6 in. wide. Find the height of the prism and the difference between its total area and the total area of the cube.

2. A pile of bricks in the form of a regular pyramid 10 ft. high is repiled in the form of a prism with an equivalent base. Assuming no loss due to piling the bricks a different way, find the height of the prism.

3. The diameter of a cylinder is 12 ft. and its height is 6 ft. Find the height of an equivalent right prism, the base of which is a square 5 ft. on a side.

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

5. The dimensions of a rectangular parallelepiped are a, b, c. Find the height of an equivalent right circular cylinder which has a for the radius of its base. Also find the height of an equivalent right circular cone which has a for the radius of its base.

6. A right circular cylinder 5 ft. in diameter is equivalent to a right circular cone 5 ft. in diameter. If the height of the cone is 6 ft., what is the height of the cylinder?

7. The heights of two equivalent right circular cylinders are in the ratio 4:9. If the diameter of the first is 5 ft., what is the diameter of the second?

8. A frustum of a cone of revolution is 5 ft. high and the diameters of its bases are 2 ft, and 4 ft. respectively. Find the height of an equivalent right circular cylinder whose radius is 5 ft.

Omit Ex. 8 if § 162 was not taken.

Exercises. Miscellaneous Problems

1. The slant height of a frustum of a regular pyramid is 24 ft., and the bases are squares whose sides are 50 ft. and 20 ft. respectively. Find the volume.

Exs. 1-6 should be omitted if §§ 125 and 162 were not taken.

2. Given that the bases of a frustum of a pyramid are regular hexagons whose sides are 2 ft. and 3 ft. respectively, and that the volume is 16 cu. ft., find the altitude.

3. From a right circular cone whose slant height is 24 ft. and the circumference of whose base is 8 ft., there is cut off by a plane parallel to the base a cone whose slant height is 5 ft. Find the lateral area and the volume of the frustum.

4. Find the difference between the volume of a frustum of a pyramid whose altitude is 8 ft. and whose bases are squares 16 ft. and 12 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 stone windmill is in the shape of a frustum of a right circular cone. The mill is 14 m. high, the outer diameters at the bottom and the top are 18 m. and 14 m., and the inner diameters are 14 m. and 12 m. respectively. How many cubic meters of stone were required to build it?

6. A brick chimney has the shape of a frustum of a regular pyramid. The chimney is 160 ft. high, its upper and lower bases are squares 9 ft. and 14 ft. on a side respectively, and a square flue 6 ft. on a side runs from top to bottom. How much brickwork does the chimney contain?

7. Two right triangles whose bases are 5 in. and 7 in., and whose hypotenuses are 83 in. and 113 in. respectively, revolve about their third sides. Find the ratio of the total areas of the solids generated and find their volumes.

BOOK VIII OF GEOMETRY

THE SPHERE

I. FUNDAMENTAL THEOREMS

163. Sphere. The locus of points in space at a given distance from a given point is called a sphere.

The terms center, radius, chord, and diameter are used as in the case of the circle.

This is the modern definition, analogous

to the modern definition of a circle, but it will be found that no confusion will arise if the student considers a sphere as the solid inclosed by a spherical surface or spheric surface. In modern mathematics the volume of a sphere is considered to mean the volume inclosed by the surface which is called a sphere.

164. A Point and a Sphere. A point may be on a sphere, within the sphere (inclosed by it), or outside the sphere (not inclosed by it).

165. Properties of a Sphere. As in $ 16, 4-7, we have

1. All radii of the same sphere or of equal spheres are equal.

2. All spheres of equal radii are equal.

3. A point is within, on, or outside a sphere according as the distance of the point from the center is less than, equal to, or greater than the radius.

There are also other properties, such as that a diameter is twice a radius, which are too obvious to require mention.

Proposition 1. Plane Intersecting a Sphere

166. Theorem. If a plane intersects a sphere, the line of intersection is a circle.

Given the plane p intersecting the sphere s with center 0. Prove that the line of intersection is a O.

Proof. Let OA, OB be any two radii of s to points on the line of intersection, and let OC be the

from 0 to p.

Then any points A and B, and hence all points on the line of intersection, lie on a O.

That is, the line of intersection is a O.

$ 16, 7

167. Great Circle. The line of intersection of a sphere and a plane passing through the center is called a great circle of the sphere.