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24. To pass planes through three given lines in space, no two of which are parallel, which shall inclose a parallelepiped.

25. Find the lateral area and the total area of a regular pyramid whose slant height is 20 in. and whose base is a square, 1 ft. on a side.

26. Find the volume of a pyramid whose altitude is 18 in. and whose base is an equilateral triangle each side of which is 8 in.

27. A regular hexagonal pyramid has an altitude of 9 ft. and each edge of the base is 6 ft. Find the volume.

28. The base of a pyramid is an isosceles triangle whose sides are 14, 25, 25, and the altitude of the pyramid is 12. Find its volume.

29. The altitude of the frustum of a pyramid is 25, and the bases are squares whose sides are 4 and 10, respectively. Find the volume of the frustum.

30. The frustum of a regular pyramid has hexagons for bases whose sides are 5 and 9, respectively. The slant height of the frustum is 14. Find its lateral area. Find its total area.

31. The altitude of a regular pyramid is 15, and each side of its square base is 16. Find the slant height, the lateral edge, the total area, and the volume.

2

2

OA2 = OD2 + DA2 = (15)2 + (8)2 = 289.

.. AO=17. OC2=OA2 + AC2=289 + 64 = 353.
.. OC √353:
=

= 18.78 +.

32. The slant height of a regular pyramid is 39, the altitude is 36, and the base is a square. Find the lateral area and volume.

33. The lateral edge of a regular pyramid is 37 and each side of the hexagonal base is 12. Find the slant height, the lateral area, total area, and volume.

In rt. A ACD, CD = 12, AC = 6, .. AD = 6 √3.
In rt. A ACO, CO = 37, AC = 6, . AO = √1333.
In rt. A CDO, CO = 37, CD = 12, .. OD = 35, etc.

34. Find the total area and volume of a regular tetrahedron whose edge is six.

The four faces are equal equilateral ▲. .. AO AC = 3 √3; :. AD=√3 and CD = 2√3.

=

Hence, OD = 2√б. Area of any face =

9√3, etc.

35. Find the total area and volume of a regular tetrahedron whose edge is 10.

36. Find the total area and volume of a regular hexahedron whose edge is 8.

37. Find the total area and volume of a regular octahedron whose edge is 16.

The 8 faces are equal equilateral ▲. A0 = 8√3. In ▲ADO, one finds OD=8√2. The volume of the octahedron = volumes of two pyramids, etc.

38. Find the total area and volume of a regular octahedron whose edge is 18.

39. The altitude of a regular pyramid is 16 and each side of the square base is 24. Find the lateral area and volume.

40. The slant height of a regular pyramid is 26 and its base is an equilateral triangle whose side is 20√3. Find the total area and volume.

41. The altitude of a regular pyramid is 29 and its base is a regular hexagon whose side is 10. Find the total area and volume.

42. Find the total area and volume of a regular tetrahedron whose edge is 18.

43. Find the total area and volume of a regular octahedron whose edge is 20.

44. If the edge of a regular tetrahedron is e, show that the total area ise2√3 and the volume is z e3 √2.

45. If the edge of a regular octahedron is e, show that the total area is 2 e2√3 and the volume is e3√2.

46. A pyramid whose base is a square 9 in. on a side, contains 360 cu. in. Find its height.

47. A pyramid has for its base a hexagon whose side is 7 units and the pyramid contains 675 cu. units. Find the altitude.

48. The volume of a regular tetrahedron is 144 √2; find its edge.

49. The volume of a regular octahedron is 243√2; find its edge.

50. The volume of a square pyramid is 676 cu. in. and the altitude is a foot. Find the side of the base. Find the lateral area.

51. The altitude of the Great Pyramid is 480 ft. and its base is 764 ft. square. It is said to have cost $10 a cu. yd. and $3 more for each sq. yd. of lateral surface (considered as planes). What was the cost?

52. The total surface of a regular tetrahedron is 324 √3 sq. in.; find its volume.

53. The base of a pyramid is a rhombus whose diagonals are 7 m. and 10 m. Find the volume if the altitude is 15 m.

54. The areas of the bases of the frustum of a pyramid are 3 sq. m. and 27 sq. m. The volume is 104 cu. m. Find the altitude.

55. The base of a pyramid is an isosceles right triangle whose hypotenuse is 8. The altitude of the pyramid is 15. Find the volume.

56. The altitude of a square pyramid, each side of whose base is 6 ft., is 10 ft. Parallel to the base and 2 ft. from the vertex a plane is passed. Find the area of the section. Find the volumes of the two pyramids concerned, and hence find the volume of the frustum.

57. Find the area of the section of a triangular pyramid each side of whose base is 8 in. and whose altitude is 18 in., made by a plane parallel to the base and a foot from the vertex.

58. The altitude of a frustum of a pyramid is 6, and the areas of the bases are 20 sq. in. and 45 sq. in. Find the altitude of the complete pyramid. Find the volume of this frustum by two distinct methods.

59. A granite monument in the form of a frustum of a pyramid, having rectangular bases one of which is 8 ft. wide and 12 ft long, and the other 6 ft. wide, is 30 ft. high. It is surmounted by a granite pyramid having the same base as the less base of the frustum, and 10 ft. in height. Find the entire volume. If one cu. ft. of water weighs 621⁄2 lb. and granite is three times as heavy as water, what is the weight of the entire monument?

60. If a square pyramid contains 40 cu. in. and its altitude is 15 in., find the side of its base.

61. A church spire in the form of a regular hexagonal pyramid whose base edge is 8 ft. and whose altitude is 75 ft. is to be painted at the rate of 18¢ per square yard. Find the cost.

62. Find the edge of a cube whose volume is equal to the volumes of two cubes whose edges are 4 and 6.

63. The base of a certain pyramid is an isosceles trapezoid whose parallel sides are 20 ft. and 30 ft. and the equal sides are each 13 ft. Find the volume of the pyramid if its altitude is 12 yards.

64. The lateral edge of the frustum of a regular square pyramid is 53 and the sides of the bases are 10 and 66. Find the altitude, the slant height, the lateral area, and the volume.

65. The sides of the base of a triangular pyramid are 33, 34, 65, and the altitude of the pyramid is 80. Find its volume.

66. The sides of the base of a tetrahedron are 17, 25, 26, and its altitude is 90. Find its volume.

67. If there are 1 cu. ft. in a bushel, what is the capacity (in bushels) of a hopper in the shape of an inverted pyramid, 12 ft. deep and 8 ft. square at the top?

68. In the corner of a cellar is a pyramidal heap of coal. The base of the heap is an isosceles right triangle whose hypotenuse is 20 ft. and the altitude of the heap is 7 ft. If there are 35 cu. ft. in a ton of coal, how many tons are there in this heap?

69. How many cubic yards of earth must be removed in digging an artificial lake 15 ft. deep, whose base is a rectangle 180 × 20 ft. and whose top is a rectangle 216 x 24 ft.? [The frustum of a pyramid.]

70. One pair of homologous edges of two similar tetrahedrons are 3 ft. and 5 ft. Find the ratio of their surfaces. Of their volumes. 71. A pair of homologous edges of two similar polyhedrons are 5 in. and 7 in. Find the ratio of their surfaces. Of their volumes.

72. The edge of a cube is 3. What is the edge of a cube twice as large? Four times as large? Half as large?

73. An edge of a tetrahedron is 6. What is the edge of a similar tetrahedron three times as large? Eight times as large? Nine times as large? One third as large?

74. An edge of a regular icosahedron is 3 in. What is the edge of a similar solid five times as large? Ten times as large? Fifty times as large? A thousand times as large?

75. The edges of a trunk are 2 ft., 3 ft., 5 ft. Another trunk is twice as long (the other edges 2 × 3 ft.). How do their volumes compare? A third trunk has each dimension double those of the first. How does its volume compare with the first? How do their surfaces compare?

76. If the altitude of a certain regular pyramid is doubled, but the base remains unchanged, how is the volume affected? If each edge

of the base is doubled, but the altitude unchanged, how is the volume affected? If the altitude and each edge of the base are all doubled, how is the volume affected?

77. If the slant height (only) of a regular pyramid is doubled, how is the lateral area affected? If each edge of the base is doubled, how is the lateral area affected? If both are doubled, what is the effect?

78. A pyramid is cut by a plane parallel to the base and bisecting the altitude. What part of the entire pyramid is the less pyramid cut away by this plane?

79. The volume of a certain pyramid, one of whose edges is 7, is 686. Find the volume of a similar pyramid whose homologous edge is 8.

80. A certain polyhedron whose shortest edge is 2 in. weighs 40 lb. What is the weight of a similar polyhedron whose shortest edge is 5 in. ? 81. An edge of a polyhedron is 5 in. and the homologous edge of a similar polyhedron is 7 in. The entire surface of the first is 250 sq. in. and its volume is 375 cu. in. Find the entire surface and volume of the second.

82. Find the edge of a cube whose volume equals that of a rectangular parallelepiped whose edges are 3 × 4 × 18.

83. A pyramid and an equivalent prism have equivalent bases. How do their altitudes compare?

84. A pyramid and a prism have the same altitude and equivalent bases. Compare their volumes.

[blocks in formation]

88. A prism whose altitude is 8 and whose base is an equilateral triangle whose side is 9 in. is transformed into a regular pyramid whose base is 10 in. square. Find its altitude.

89. An altitude of a pyramid is 10 m. How far from the vertex will a plane parallel to the base divide the pyramid into two equivalent parts?

90. The altitude of a pyramid is 12, and two planes are passed parallel to the base and dividing the pyramid into three equivalent parts. At what distances from the vertex are they?

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