PROBLEMS OF COMPUTATION

1. What is the length of the longest line that can be drawn within a rectangular parallelepiped 12 ft. long, 4 ft. wide, and 3 ft. thick?

2. Find the total area of a rectangular parallelepiped whose dimensions are 14 in., 11 in., and 8 in.

3. If the diagonal of a cube is 24 in., what is the total area of the cube? What is the volume?

4. The area of the entire surface of a cube is S. Find the length of an edge, the length of a diagonal, and the volume.

5. Find the number of square inches on the surface of a cube whose volume is 427 cu. in.

6. If 180 sq. ft. of lead are required for lining the bottom and sides of a cubical vessel, how many cubic feet of water will it hold?

7. The number of square feet on the surface of a certain cube is equal to the number of cubic feet in its volume. Find the length of an edge.

8. A rectangular parallelepiped has a square base whose side is 23 in., and the diagonal of the parallelepiped is 52 in. Find the volume. 9. The dimensions of the base of a rectangular parallelepiped are 12 ft. and 4 ft., and the number of square feet in the total area is equal to the number of cubic feet in the volume. Find the altitude.

10. In a rectangular parallelepiped whose altitude is 8 in. and the total area is 160 sq. in.

base is a square, the Find the volume.

11. The three external dimensions of a box with cover are 2 ft. 8 in., 1 ft. 10 in., and 1 ft. 6 in., and it is constructed of a material 1 in. thick. Find the number of cubic inches of material used.

12. A cube whose edge measures 15 in. is equivalent to a rectangular parallelepiped, two of whose dimensions are 25 in. and 10 in. Find the third dimension and the total area of the parallelepiped.

13. Three edges of a parallelepiped are AB, AC, and AD. Find the ratio of the volumes of the two solids into which the parallelepiped is divided by the plane BCD. Find the ratio of the two solids into which the parallelepiped is divided by the plane bisecting these three edges.

14. The base of a right prism is a triangle whose sides are 12 in., 15 in., and 17 in., and the altitude is 8 in. Find the lateral area.

15. Find the volume of a triangular prism, the sides of whose base are 8 in., 6 in., and 5 in., and whose altitude is 10 in.

16. The volume of a right prism is 480 cu. in., and its base is an isosceles triangle whose sides are 13 in., 13 in., and 10 in. Find the altitude of the prism.

17. The lateral area and the volume of a regular triangular prism are 144 sq. in. and 96 √3 cu. in. respectively, Find the altitude and an edge of the base.

18. A right prism 16 in. high has for its base a regular hexagon whose side is 12 in. Find the total area.

19. Find the volume of a regular hexagonal prism whose altitude is 6 ft., and each side of whose base is 4 ft.

20. Find the number of cubic feet of concrete in a dam 250 ft. long, 31 ft. high, 33 ft. wide at the bottom, and 5 ft. wide at the top.

21. The base of a truncated right prism is an isosceles triangle whose sides are 5 in., 5 in., and 6 in., and the lateral edges are 10 in., 13 in., and 16 in. Find the volume.

22. A right section of a truncated prism is an equilateral triangle whose perimeter is 16 in., and the lateral edges are 8 in., 10 in., and 10 in. Find the volume.

23. The base of a truncated right prism is a square, each side of which is 6 in., and the lateral edges are 5 in., 8 in., 13 in., and 10 in. Find the volume.

24. Find the volume of a truncated triangular prism if its base contains 40 sq. in., the three lateral edges are 5 in., 10 in., and 20 in., and the projection upon the base of the edge of length 5 in. equals 4 in.

25. Find the lateral edge, lateral area, and volume of a regular pyramid whose altitude is 10 in., and whose base is an equilateral triangle each of whose sides is 8 in.

26. Find the lateral area and volume of a regular pyramid whose slant height is 12 ft., and whose base is an equilateral triangle each of whose sides is 5 √3 ft.

27. Find the lateral edge, lateral area, and volume of a regular quadrangular pyramid, each side of whose base is 10 in., and whose altitude is 12 in.

28. Find the lateral area and volume of a regular hexagonal pyramid, each side of whose base is 4 in., and whose altitude is 5 in.

29. The volume of a pyramid is 210 cu. in., and its base is a triangle whose sides are 5 in., 12 in., and 13 in. Find the altitude of the pyramid.

30. The lateral surface of a regular quadrangular pyramid is composed of four equilateral triangles, and its altitude is 15 in. Find the area of the base.

31.

A regular pyramid has a lateral edge of 101 ft. and a square base 40 ft. on a side. Find the volume.

32. The lateral area square whose edge is a.

of a regular pyramid is S, and the base is a Find the altitude of the pyramid.

33. Find the volume of a regular hexagonal pyramid whose altitude and slant height are 24 in. and 25 in. respectively.

34. The altitude of a regular hexagonal pyramid is 15 ft., and the apothem of the base is 8 ft. Find the lateral area of the pyramid.

35. The altitude of a regular hexagonal pyramid is 5 ft., and an edge of the base is 4 ft. Find the lateral area of the pyramid.

36. The base of a regular pyramid is a hexagon of side 10 in. The lateral edge is 20 in. Find the volume.

37. The base of a regular pyramid is a regular hexagon which can be inscribed in a circle of radius 10 in. A lateral edge of the pyramid is 20 in. Find the volume.

38. The base of a regular pyramid is a hexagon whose side is 4 ft., and its lateral area is six times the area of the base. Find the altitude of the pyramid.

39. The spire of a church is a regular hexagonal pyramid; each side of the base is 10 ft., and the height is 50 ft. There is also a hollow part which is also a regular hexagonal pyramid; the height of the hollow part is 45 ft., and each side of the base is 9 ft. Find the number of cubic feet of stone in the tower.

40. One lateral edge of a pyramid is 10 in., and the angle formed by this edge and the plane of the base is 45°; the area of the base is 120 sq. in. Find the volume.

41. A regular prism and a regular pyramid have equivalent volumes and equivalent bases. If the altitude of the prism is 71⁄2 ft., what is the altitude of the pyramid?

42. The areas of two mutually perpendicular faces of a tetrahedron are 4 sq. in. and 3 sq. in. The edge between these two faces is 2 in. long. Find the volume of the tetrahedron.

43. The vertices of a tetrahedron are four vertices of a cube, no two of which lie on the same edge of the cube. Find the ratio of the volume of the tetrahedron to the volume of the cube.

44. A pedestal for a statue is in the form of the frustum of a regular quadrangular pyramid. Each side of the lower base is 5 ft., each side of the upper base is 3 ft., and the height is 4 ft. Find the lateral area of the pedestal.

45. The lower base of a frustum of a right pyramid is a square 4 in. on a side. A side of the upper base is half that of the lower base, and the altitude of the frustum is the same as a side of the upper base. Find the volume of the frustum.

46. Find the lateral area and volume of a frustum of a regular quadrangular pyramid, the sides of whose bases are 20 ft. and 4 ft., and whose altitude is 15 ft.

47. The slant height of the frustum of a regular quadrangular pyramid is 37 in., and the sides of the lower base and upper base are 42 in. and 18 in. respectively. Find the volume.

48. Find the volume of the frustum of a triangular pyramid whose bases are 90 sq. in. and 40 sq. in., and whose altitude is 3 in.

49. The volume of a frustum of a pyramid is 84 cu. in., and the bases are squares whose sides are 2 in. and 4 in. Find the altitude.

50. If the bases of a frustum of a pyramid are two regular hexagons whose sides are 1 ft. and 2 ft., and the volume of the frustum is 12 cu. ft., find the altitude.

51. A stick of timber 32 ft. long and 18 in. wide is 15 in. thick at one end and 12 in. thick at the other. Find the number of cubic feet it contains.

52. The volume of a frustum of a pyramid is 38 cu. ft., its altitude is 6 ft., and the area of one base is 9 sq. ft. Find the area of the other base.

53. Find the volume of the frustum of an oblique pyramid from following data: the lower base is a square 4 in. on a side, the upper base is a square 2 in. on a side, and one of the inclined edges, which is 8 in. long, has as its projection upon the lower base one of the diagonals of that base.

54. Find the total surface and volume of a regular tetrahedron whose edge is 8 in.

55. Find the volume of a regular tetrahedron whose altitude is 7 in. 16

56. The volume of a regular tetrahedron is √2 cu. in. Find 3 the edge, slant height, and altitude.

57. Find the length of an edge of a regular tetrahedron whose volume is 18 √2 cu. in.

58. Find the volume of a regular tetrahedron whose total area is 144 √3 sq. in.

59. The vertices of one regular tetrahedron are at the centers of the faces of another regular tetrahedron. Find the ratio of the volumes. 60. Find the sum of the face angles of the polyhedral angle at any vertex of a regular octahedron.

61. The area of one face of a regular octahedron is one square foot. Find the volume.

62. Find the volume of a regular octahedron whose total area is 72 √3 sq. in.

63. Find the ratio of the volume of a cube to that of the regular octahedron whose vertices are the centres of the faces of the cube.

64. The corner of a cube is cut off by a plane passing through the mid-points of the edges which terminate at that vertex, and the process is repeated for each corner of the cube. Find the ratio of the volume of the solid that remains to the volume of the cube.

65. AB, AC, and AD are three edges of a cube which meet in the vertex A. A plane is passed through the mid-points of these edges. If the cube contains 8 cu. ft., find the volume of the corner cut off by the plane and the length of the perpendicular drawn from the centre of the cube to the plane.

66. A pyramid 12 ft. high has a base containing 225 sq. ft. What is the area of a section parallel to the base whose distance from the vertex is 8 ft.?

67. A pyramid whose base is a square 5 in. on an edge is cut by a plane parallel to the base and bisecting the altitude. Find the area of the section.

68. A pyramid 9 ft. high has a square base measuring 6 ft. on a side, and a section parallel to the base measures 2 ft. on a side. Find the distance from the vertex to this section.