42. If water sufficient to fill a cistern in 50 minutes will pass through a pipe 2 inches in diameter, how long will it take to fill the cistern if the pipe is 5 inches in diameter, no allowance being made for friction?

43. If through a pipe 3 inches in diameter a vat can be filled in 4 hours and 16 minutes, what must be the diameter of a pipe through which it can be filled in 49 minutes, no allowance being made for friction?

44. If it costs $50 to paint a house 45 feet in length, what ought it to cost to paint a similar house 25 feet in length?

45. If a pipe 2 inches in diameter delivers 217 quarts of water a minute, how many quarts will a pipe 9 inches in diameter deliver a minute?

46. If a pipe 1.5 inches in diameter discharges 10 gallons a minute, how much will a pipe 3.5 inches in diameter discharge in the same time?

47. If we increase the length of the rope with which a horse is tied to a post in the centre of a field, from 12 to 15 feet, allowing 3 feet that the horse can reach beyond the end of the rope, how much have we increased the area over which he can feed?

48. If it costs $ 45 to fence a field containing 10 acres, how much will it cost to enclose, with the same kind of fence, a field of similar shape containing 8 acres?

SOLIDS.

485. A Prism is a solid that has two equal, parallel faces, called bases, and all its other faces parallelograms.

NOTE. A prism is triangular, quadrangular, pentagonal, etc., according as its bases are triangles, quadrangles, pentagons, etc.

486. A round body whose diameter is the same throughout its entire length, and whose ends or bases are equal, parallel circles, is a Cylinder.

487. The altitude of a prism or cylinder is the perpendicular distance between its bases.

488. The surface of a prism or cylinder can be found by multiplying the perimeter or circumference of the base by the length of the solid, and to the product adding the area of the two ends.

49. What is the surface of a prism whose length is 20 inches and base 6 inches square? Ans. 552 sq. in.

50. What is the surface of a cylinder whose length is 30 yards and diameter 2 yards?

489. The solid contents of a prism or cylinder is equal to the product of the area of its base and its altitude.

51. What are the contents of a cylinder whose length is 25 feet and diameter 15 feet?

52. What are the contents of a quadrangular prism whose length is 15 feet and base a rectangle 3 by 4 feet?

490. A Pyramid is a solid, having a polygon for its base, and for its other faces triangles which meet at a common point, called the vertex of the pyramid.

491. The slant height of a pyramid is the perpendicular distance from the vertex to one side of the base.

NOTE. A pyramid is triangular, quadrangular, etc., according as its base is a triangle, quadrangle, etc.

492. A solid that is like a pyramid, except that its base is a circle, is a Cone.

493. The altitude of a pyramid or cone is the perpendicular from the vertex to the base.

494. The contents of a pyramid or of a cone can be found by multiplying the area of its base by one third of its altitude.

53. What are the contents of a cone whose base is 15 inches in diameter and whose altitude is 42 inches?

54. What are the contents of a pyramid whose altitude is 18 inches, and whose base is a triangle having sides 4, 6, and 8 inches, respectively?

495. A Frustum of a pyramid or cone is the part remaining after a portion next the vertex has been cut off by a plane parallel to the base. The two ends are called the upper and lower bases.

496. The contents of a frustum of a pyramid or cone can be found by multiplying the sum of the areas of the two bases, added to the mean proportional between the two bases, by one third of the altitude of the frustum.

55. What are the contents of the frustum of a quadrangular pyramid, whose altitude is 12 feet and whose bases are 6 feet and 4 feet square? Ans. 304 cu. ft.

56. What are the contents of the frustum of a cone whose altitude is 42 inches and whose bases are 24 inches and 18 inches in diameter ?

57. If the inner diameter of the bottom of a tub is 21 inches, and of the top 24 inches, and the height is 12 inches, how many gallons will it hold?

497. A Sphere is a solid bounded by a curved surface, of which every point is equally distant from a point within, called the centre.

The diameter of a sphere is a line passing through the centre, and limited in both directions by the surface.

498. The surface of a sphere is equal to the product of its circumference and its diameter.

Surface of a Sphere = Circumference x Diameter.

58. What is the surface of a sphere whose diameter is 100 inches? Ans. 31416 sq. in.

59. What is the surface of a cannon ball whose diameter is 6 inches?

60. What is the surface of a globe whose diameter is 2 feet?

499. The contents of a sphere is equal to the product of its surface and one third of its radius. Or it is equal to 0.5236 (of 3.1416) times the cube of its diameter.

Volume of a Sphere = Surface

Radius.

= 0.5236 × Diameter.

61. What are the contents of a sphere whose diameter is 100 inches? Ans. 523600 cu. in.

62. What is the volume of a cannon ball whose diameter is 8 inches?

63. What is the volume of a globe whose diameter is 3 feet? 64. What is the volume of a ball whose radius is 2 inches?

500.

SIMILAR SOLIDS.

Similar Solids are solids which are of precisely the same form.

The solid contents of similar solids are to each other as the cubes of their corresponding lines; and, conversely, the corresponding lines of similar solids are to each other as the cube roots of their contents.

65. If a cannon ball 8 inches in diameter weighs 64 pounds, what is the weight of a cannon ball 4 inches in diameter ?

88 : 43 = 64 lb. : Ans., or 28 : 13 = 64 lb. : Ans. ;

that is, 8:1 = 64 lb. : 8 lb., Ans.

66. If a sphere of silver 24 inches in diameter is worth $64, what is the diameter of a sphere of silver that is worth $1728?

3

3

3

=

64: 1728 = 24 in. Ans., or 4: 12 24 in. Ans.

Ans. 7 in.

67. Of two eggs of exactly the same shape, the longest axis of one is 23 inches, and of the other 2 inches. Compare their volumes.

68. If two men own a conical stack of hay 15 feet in height, what must be the altitude of the frustum left after one has taken half of the hay from the top?

69. If a pail whose height is 22 inches holds 10 quarts, what must be the height of a similar pail to hold 12 quarts?

70. If a horse whose girth is 6 feet weighs 950 pounds, what will a horse whose girth is 7 feet weigh?

71. If the altitude of a box holding a bushel is 20 inches, what is the altitude of a box of similar shape holding 19.2 quarts?