679. Cor. IV. The volumes of two spheres are to each other as the cubes of their radii, or as the cubes of their diameters.

(The proof is left to the pupil.)

680. Cor. V. The volume of a spherical pyramid is equal to the area of its base multiplied by one-third the radius of the sphere.

Given P the volume of a spherical pyramid, K the area of its base, and R the radius of the sphere.

P=K× R.

To Prove Proof. Let n denote the number of sides of the base of the spherical pyramid, s the sum of its referred to a rt. as the unit of measure, T the area of a tri-rectangular A, T" the volume of a tri-rectangular pyramid, S the area of the surface of the sphere, and V its volume.

681. Given the radii of the bases, and the altitude, of a

spherical segment, to find its volume.

Given the centre of arc ADB, lines AA' and BB' to diameter OM, AA' r', BB' = r, A'B' = h, and figure ADBB'A' revolved about OM as an axis.

=

Required to express volume of spherical segment generated by ADBB'A' in terms of r, r', and h.

Solution. Draw lines OA, OB, and AB; also, line OCL AB, and line AE 1 BB'; and denote radius OA by R.

Now, vol. ADBB'A' = vol. ACBD + vol. ABB'A'.

vol. ACBD = vol. OADB – vol. OAB.

(1)

Also,

Now,

Also,

= { AB ̊.

.. vol. ACDB = } π × ‡ AB2 × h = { πAB2 h.

Substituting in (1), we have

vol. ADBB'A'

(?)

(?)

(§ 661)

= } π [(r − r')2 + h2] h + } # (2r22 + 2p12 + 2rr') h

12

= { π (y2 — 2rr' + po12 + h2 + 2 p2 + 2p12 + 2rr') h

= {(3r2 + 3r12) h + } #h3

π

682. Cor. If r denotes the radius of the base, and h the altitude, of a spherical segment of one base, its volume is

} πr2h + } #h3.

EXERCISES.

25. Find the volume of a sphere whose radius is 12.

26. Find the volume of a spherical sector, the altitude of whose base is 12, the diameter of the sphere being 25.

27. Find the volume of a spherical segment, the radii of whose bases are 4 and 5, and whose altitude is 9.

28. Find the radius and volume of a sphere, the area of whose surface is 324 π.

29. Find the diameter and area of the surface of a sphere whose volume is 1125 π.

30. The surface of a sphere is equivalent to the lateral surface of its circumscribed cylinder.

31. The volume of a sphere is two-thirds the volume of its circumscribed cylinder.

32. A spherical cannon-ball 9 in. in diameter is dropped into a cubical box filled with water, whose depth is 9 in. How many cubic inches of water will be left in the box? (T = 3.1416.)

33. What is the angle of the base of a spherical wedge whose volume is 40T, if the radius of the sphere is 4?

34. Find the volume of a quadrangular spherical pyramid, the angles of whose base are 107°, 118°, 134°, and 146°; the diameter of the sphere being 12.

35. The surface of a sphere is equivalent to two-thirds the entire surface of its circumscribed cylinder.

36. Prove Prop. IX. when the straight line is parallel to the axis. 37. Find the area of the surface and the volume of a sphere inscribed in a cube the area of whose surface is 486.

38. How many spherical bullets, each in. in diameter, can be formed from five pieces of lead, each in the form of a cone of revolution, the radius of whose base is 5 in., and whose altitude is 8 in. ?

39. A cylindrical vessel, 8 in. in diameter, is filled to the brim with water. A ball is immersed in it, displacing water to the depth of 24 in. Find the diameter of the ball.

40. If a sphere 6 in. in diameter weighs 351 ounces, what is the weight of a sphere of the same material whose diameter is 10 in. ?

41. If a sphere whose radius is 12 in. weighs 3125 lb., what is the radius of a sphere of the same material whose weight is 819 lb. ?

42. The altitude of a frustum of a cone of revolution is 31, and the radii of its bases are 5 and 3; what is the diameter of an equivalent sphere ?

43. Find the radius of a sphere whose surface is equivalent to the entire surface of a cylinder of revolution, whose altitude is 101, and radius of base 3.

44. The volume of a cylinder of revolution is equal to the area of its generating rectangle, multiplied by the circumference of a circle whose radius is the distance to the axis from the centre of the rectangle.

45. The volume of a cone of revolution is equal to its lateral area, multiplied by one-third the perpendicular from the vertex of the right angle to the hypotenuse of the generating triangle.

46. Two zones on the same sphere, or equal spheres, are to each other as their altitudes.

47. The area of a zone of one base is equal to the area of the circle whose radius is the chord of its generating arc. (§ 270, 2.)

48. If the radius of a sphere is R, what is the area of a zone of one base, whose generating arc is 45° ? (Ex. 55, p. 210.)

49. If the altitude of a cone of revolution is 15, and its slant height 17, find the total area of an inscribed cylinder, the radius of whose base is 5.

(Let the cone and cylinder be generated by the revolution of rt. A ABC and rect. CDEF about AC as an axis.)

50. Find the area of the surface and the volume of a sphere circumscribing a cylinder of revolution, the radius of whose base is 9, and whose altitude is 24.

51. An equilateral triangle, whose side is 6, revolves about one of its sides as an axis. Find the area of the entire surface, and the volume, of the solid generated.

52. A cone of revolution is inscribed in a sphere whose diameter is the altitude of the cone. Prove that its lateral surface and vol

ume are, respectively,

and

the surface and volume of the sphere.

53. Find the volume of a sphere circumscribing a cube whose volume is 64.

54. A cone of revolution is circumscribed about a sphere whose diameter is two-thirds the altitude of the cone. Prove that its lateral surface and volume are, respectively, three-halves and nine-fourths the surface and volume of the sphere.

55. If the radius of a sphere is 25, find the lateral area and volume of an inscribed cone, the radius of whose base is 24.

C

D'

(Two solutions.)

B

56. If the volume of a sphere is 500, find the lateral area and volume of a circumscribed cone whose altitude is 18.

57. Find the volume of a spherical segment of one base whose altitude is 6, the diameter of the sphere being 30.

58. A square whose area is A revolves about its diago

nal as an axis. Find the area of the entire surface, and c the volume, of the solid generated.

B

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59. The altitude of a cone of revolution is 9. At what distances from the vertex must it be cut by planes parallel to its base, in order that it may be divided into three equivalent parts? (§ 656.)

(Let V denote the volume of the cone, x the distance from the vertex to the nearer plane, and y the distance to the other.)

60. Given the radius of the base, R, and the total area, T, of a cylinder of revolution, to find its volume.

(Find H from the equation T=2πRH+2π R2.)

61. Given the diameter of the base, D, and the volume, V, of a cylinder of revolution, to find its lateral area and total area.

62. Given the altitude, H, and the volume, V, of a cone of revolution, to find its lateral area.

63. Given the slant height, L, and the lateral area, S, of a cone of revolution, to find its volume.