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Exercises. Cylinders

1. The volume of a right circular cylinder is the product of the lateral area and half the radius.

2. The volume of a right circular cylinder is the product of the area of the rectangle, which generates the cylinder by revolving about one of its sides, multiplied by the circumference of the circle generated by the point of intersection of the diagonals of the rectangle.

3. If the altitude of a right circular cylinder is equal to the diameter of the base, the volume is the product of the total area and one third the radius.

4. By what number must the dimensions of a cylinder of revolution be multiplied to obtain a similar cylinder of revolution (Ex. 9, page 102) whose entire surface area is twice the first? n times the first?

5. What is the multiplier in Ex. 4 if the volume of the second cylinder is to be twice that of the first? is to be n times that of the first?

6. Compare the volumes of the solids generated by the successive revolution of a rectangle of base b and altitude h about two adjacent sides.

7. Find the radius of a right circular cylinder in which the number of cubic units of volume is equal to the number of square units of the area of the entire surface.

8. Find to the nearest square centimeter the area of the total surface of a cylinder whose altitude is 7.6 cm. and the diameter of whose base is 4.2 cm.

9. Find to the nearest cubic centimeter the volume of a cylinder whose altitude is 6 cm. and which fits exactly into a right prism whose base is a square that is 1.8 cm. on a side.

Exercises. Cones and Pyramids

1. The altitude of a cone of revolution is 24 in. and the radius of the base is 10 in. Find the radius of the sector of paper which, when rolled up, will just cover the curve surface of the cone. Find also the number of degrees in the angle of this sector.

The second result may be expressed either in degrees with a decimal fraction, or in degrees, minutes, and seconds.

2. The volume of a regular pyramid is the product of one third its lateral area and the perpendicular distance from the center of the base to any lateral face.

3. Find the volume of a pyramid whose base is 30 sq. in. and one of whose lateral edges, which makes an angle of 45° with the plane of the base, is 5 in. long.

4. A pyramid is cut by a plane parallel to the base and bisecting the altitude. What is the ratio of the volume of the pyramid cut off to that of the entire pyramid?

5. Consider Ex. 4 for the case of a cone.

6. The height of a regular hexagonal pyramid is 6 in. and one edge of the base is 1 in. Find the volume and also find the volume of the pyramid cut off by a plane 4 in. from the base and parallel to it.

7. One of the lateral edges of a regular hexagonal pyramid is 6 in., and the radius of the circle circumscribed Find the altitude, the volume, the

about the base is 1 in.

lateral area, and the area of the total surface.

8. If a right triangle of hypotenuse h and sides a and b revolves about h as an axis, what is the volume of the solid thus generated?

9. If the radius of the base of a right circular cone is r and the angle at the vertex is 120°, what is the volume?

Exercises. Spheres, Cylinders, and Cones

1. The area of a sphere is two thirds the area of the total surface of the circumscribed cylinder.

2. The volume of a sphere is two thirds the volume of the circumscribed cylinder.

Exs. 1 and 2 were discovered by Archimedes, one of the greatest mathematicians of Greece, about 250 B.C.

3. A sphere of radius 6 in. and a right circular cone of the same radius stand on a plane. If the height of the cone is equal to the diameter of the sphere, find the position of the plane that cuts the two solids in equal circular sections.

4. In a cylindric jar 8 in. in diameter, water is standing to a depth of 6 in. If an iron ball 4 in. in diameter is dropped into the jar, what is then the depth of the water?

5. On the base of a right circular cone a hemisphere is constructed outside the cone. Given that the area of the hemisphere is equal to that of the cone, and that the radius is r, find the slant height of the cone, the inclination of the slant height to the base, and the volume of the entire solid. 6. Find the area of a sphere inscribed in a cylinder of volume 4 πα.

7. Find the volume of a sphere inscribed in a cylinder of area πd (2+d).

8. A sphere of radius r is inscribed in a cylinder. Find the volume of the cylinder not occupied by the sphere.

9. A cylinder is circumscribed about a hemisphere, and a cone is inscribed in the cylinder so as to have its vertex on the upper base and to have its base in common with the lower base of the cylinder. Prove that the volumes of the cone, the hemisphere, and the cylinder are proportional to 1, 2, 3.

Exercises. Portions of the Surface of a Sphere

1. If the altitude of the north temperate zone is 1800 mi., what is the area of the zone?

In Exs. 1-7 take 4000 mi. as the radius of the earth.

2. How far in one direction can a man see from an ocean steamer if his eye is 50 ft. above the water?

3. How many square miles of the earth's surface can be seen from an airplane at an elevation of 10,000 ft.?

4. At what height above the earth must a man be in order to see one eighth of the surface?

5. What fractional part of the earth's surface could be seen if an observer were at the height of the earth's radius above the sea?

6. If the ocean area is three fourths of the earth's surface and the average depth of the water is 2 mi., what is the volume of water in the oceans?

7. In a lighthouse on an isolated rock the light is placed 168 ft. above the surface of the sea and can be seen from any point within a circle reaching to the horizon. Find the number of square miles of the earth's surface inclosed by this circle.

Find the areas of triangles with angles as follows on spheres of the given radii:

8. 120°, 110°, 90°; r = 7 in.

9. 105°, 105°, 80°; r = 7 in.

10. 120°, 95°, 90°; r = 14 in.

11. 90°, 90°, 90°; r = 91 in.

Find the areas of polygons with angles as follows on spheres of the given radii:

12. 140°, 150°, 80°, 80°; r = 14 in.

13. 120°, 100°, 185°, 80°, 100°; r = 21 in.

Exercises. Spherical Polygons and Polyhedral Angles

1. The planes which are perpendicular to the three faces of a trihedral angle and bisect the face angles meet in a straight line. After proving the proposition state the corresponding one relating to a spherical triangle.

It is unnecessary to prove the latter, since it follows by § 193.

2. The planes passing through the edges of a trihedral angle and perpendicular to the opposite faces meet in a straight line. Consider also, as in Ex. 1, the corresponding proposition relating to a spherical triangle.

3. Find the area of a spherical triangle, given that the perimeter of its polar triangle is 297° and that the radius of the sphere is 10 in.

4. Find the spherical excess of a spherical triangle whose angles are 87°, 92°, and 106°; of a spherical quadrilateral whose angles are 145°, 92°, 75°, and 125°.

5. If the two polygons of Ex. 4 are both on a sphere of radius 10 in., what is the area of each?

On a sphere of radius 7 in., find the areas of spherical triangles with angles as follows:

[blocks in formation]

On a sphere of radius 14 in., find the areas of spherical polygons with angles as follows:

12. 80°, 90°, 100°, 110°.

14. 96°, 72°, 116°, 130°.

13. 72°, 88°, 110°, 120°.

15. 100°, 100°, 100°, 100°.

16. Discuss the case of the area of a spherical triangle whose angles are 200°, 280°, and 60° on a sphere whose radius is 10 in.

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