Notation:

Notation:

1. S (2 r. h) = r. h = 3.1416. r. h.

=

1. S(r+ r') h

S'

=

radius of upper base;

volume (of any frustum).

3.1416 (rr') h.

2. S′ = = [(r + 2o′ ) π + po2 + po2 2 ] = 3.1416 [(r + r2 ) π + po2 + 2o2].

The student should be careful to distinguish between the decimal point and the sign

of multiplication.

1. The volumes of two spheres are proportional to the cubes of their radii or to the cubes of their diameters.

2. The surface of a sphere is to the entire surface of the circumscribed cylinder as the volume of the sphere is to the volume of the cylinder. Suggestion.-Surface of cylinder, entire, equals 6 π r2.

Volume of cylinder = § π μ3.

Surface and volume of sphere = what?

(?) (?)

3. The radius of a great circle of a sphere is 4; what is the surface of

the sphere? (See Th. III, Cor. 1.)

4. The surface of a sphere is 416; what is its radius?

5. The radii of two spheres are 6 and 16 respectively; what is the ratio of their surfaces?

6. The radius of a sphere is 10; what is the entire area of the circumscribing cylinder?

7. The diameter of a sphere is 10: circle at a distance 4 from the centre.

required the circumference of a small (See Th. I, Cor. II.)

8. The diameter of a sphere is 36; required its volume.

9. The surface of a sphere is 804.252; what is its volume?

10. The altitude of a cylinder is 18 inches, and the diameter of its base 6 inches; required the volume and radius of the inscribed sphere.

11. The area of a small circle of a sphere is 314.16, and is situated at a distance 3 from the centre of the sphere; required the volume of the sphere.

12. From any point in the surface of a hemisphere there can be two perpendiculars drawn to the circumference of the great circle which forms its base, and they will in general be unequal; one measuring the least distance to that circumference, and the other the greatest.

13. Two oblique arcs drawn from the same point to points of this circumference at equal distances from the foot of the perpendicular, are equal.

14. Of two oblique arcs, that is the longer which meets this circumference at the greater distance from the foot of the perpendicular.

15. Required the side of a cube that can be B cut out of a sphere whose diameter is 2 feet.

Suggestion.-Let A B be the cube; then CE is

the diameter of the sphere.

Find ED, remembering that EDC is a right triangle, as also A D C; A C, A D, and E D being equal to each other.

d

D

A

16. A stone, of irregular shape, was immersed in a cylindrical vessel 3 feet in diameter, and was observed to raise the level of the water 10 inches; what was the volume of the stone?

17. What must be the inside dimensions of a cubical box, to hold 400 balls, each 34 inches in diameter?

18. The volume and surface of a sphere are expressed by the same numbers; required its diameter.

19. The height of a mountain was measured, and found to be 4 miles; the distance A B at which it could be seen, was found to be 179 miles; required the volume of the earth.

20. What is the locus of the points in space at a given distance from a fixed point?

21. The surface of a sphere is 3;

what is the surface of a sphere whose volume is 4 times as great?

B

Dk

266

MISCELLANEOUS EXERCISES.

22. The volume of a cone is c; the radius of its base is r; required the volume of the circumscribed sphere.

28. With a given radius, to describe three equal circles which shall touch one another, and then to describe another circle which shall touch them all three.

29. The length, width, and height of a room are 6, 3, and 4; required its diagonal, A B.

30. Find the length of an arc of 28° on a circumference whose radius is 10.

31. Find the length of an arc of 45°

45', the radius being 15.

32. Find the area of a sector whose

arc is 30° 30', and whose radius is 24.

6

33. The entire surface of a cube is a; required its volume, and its diag

onal.

34. The area of an equilateral triangle is equal to one-fourth of the square of one side, multiplied by the square root of 3.

35. Find the entire surface of a right prism whose altitude is 16, and whose base is an equilateral triangle, each side of which is 6.

36. Find the entire surface of a right prism whose altitude is 18, and whose base is a regular hexagon, each side of which is 5.

37. Find the entire surface of a right pyramid whose slant height is 16 feet, and whose base is three feet square.