Find to the nearest gallon the quantity of water which will be held by cylindrical vessels having the following dimensions: 21. Radius of base 10 inches; height 20 inches. 22. Radius of base 2 feet 6 inches; height 4 feet. 24. Radius of base 7 feet 6 inches; height 10 feet. 25. The height of a prism is 24 feet; the base is a trapezoid, the parallel sides being 18 feet and 12 feet respectively, and the distance between them 5 feet: find the volume. 26. The wall of China is 1500 miles long, 20 feet high, 15 feet wide at the top, and 25 feet wide at the bottom: find how many cubic yards of material it contains. 27. Find the number of cubic feet of earth which must be dug out to form a ditch 1000 feet long, 8 feet deep, 16 feet broad at the bottom, and 20 feet broad at the top. 28. Find to the nearest gallon the quantity of water which will be required to fill a ditch having the following dimensions: length 40 feet, depth 6 feet, breadth at the top 10 feet, breadth at the bottom 8 feet. 29. A ditch is 8 feet deep, 24 feet broad at the top and 16 feet broad at the bottom: find the length of the ditch if 250000 cubic feet of earth are dug out to make it. 30. A ditch is 4 feet deep, 5 feet broad at the top, and 4 feet broad at the bottom: find the length of the ditch, to the nearest foot, if it will hold 10000 gallons of water. 31. Find how many cubic feet of earth must be dug out to make a well 3 feet in diameter and 30 feet deep. 32. Find how many cubic yards of earth must be dug out to make a well 4 feet in diameter and 119 feet deep. 33. Find the number of cubic yards of earth dug out of a tunnel 100 yards long, whose section is a semicircle, with a radius of 10 feet. 34. Find how many pieces of money of an inch in diameter, and of an inch thick, must be melted down in order to form a cube whose edge is 3 inches long. 35. The diameter of a well is 4 feet, and its depth 30 feet: find the cost of excavation at 7s. 6d. per cubic yard. 36. The diameter of a well is 3 feet 6 inches, and its depth 40 feet: find the cost of excavation at 7s. 6d. per cubic yard. 37. The diameter of a well is 3 feet 9 inches, and its depth 45 feet: find the cost of excavation at 7s. 3d. per cubic yard. 38. If 30 inches of gunpowder weigh a pound, find what length of a gun, 6 inches bore, will be filled with 10 lbs. of powder. 39. A cubic foot of brass is to be drawn into a wire of an inch in diameter: find the length of the wire. 40. A cubic foot of brass is to be drawn into a wire 025 of an inch thick: find the length of the wire. 41. Find the volume of a cylindrical shell, the radius of the inner surface being 5 inches, that of the outer 6 inches, and the height 7 feet. 42. Find the volume of a cylindrical shell, the radius of the outer surface being 10 inches, the thickness 2 inches, and the height 9 feet. 43. Find the volume of a cylindrical shell, the radius of the inner surface being 12 inches, the thickness 3 inches, and the height 10 feet. 44. An iron pipe is 3 inches in bore, half an inch thick, and 20 feet long: find its weight, supposing that a cubic inch of iron weighs 4'526 ounces. 45. The length of a leaden pipe is 13 feet, its bore is 1 inches, and its thickness 1 inches: finds its weight, supposing a cubic inch of lead to weigh 6·604 ounces. 46. Find the cost of a leaden pipe of 2 inches bore, which is half an inch thick and 8 yards long, at 23d. per lb., supposing a cubic foot of lead to weigh 11412 ounces. 47. A square iron rod, an inch thick, weighs 10 lbs. : find the weight of a round iron rod of the same length and thickness. 48. Every edge of a certain triangular prism measures 10 inches: find the volume. 49. The base of a certain prism is a regular hexagon; every edge of the prism measures 1 foot: find the volume of the prism. 50. The radius of the inner surface of a leaden pipe is 11⁄2 inches, and that of the outer is 1 inches: if the pipe be melted, and formed into a solid cylinder of the same length as before, find the radius.. 51. The trunk of a tree is a right circular cylinder, 3 feet in diameter and 20 feet high: find the volume of the timber which remains when the trunk is trimmed just enough to reduce it to a rectangular parallelepiped on a square base. The following examples involve the extraction of the cube root: 52. The sides of the base of a triangular prism are 52, 51, and 25 inches respectively; and the height is 60 inches: find the length of a cube of equivalent volume. 53. The height of a cylinder is 4 feet 9 inches, the radius of the base is 4 feet 3 inches: find the length of a cube of equivalent volume. 54. Suppose a sovereign to be of an inch in diameter, and of an inch in thickness; if 100000 of them be melted down and formed into a cube, find the length of the cube. 55. The height of a cylinder is to be 10 times the radius of the base, and the volume is to be 25 cubic feet: find the radius. 56. The height of a cylindrical vessel is to be half the radius of the base, and the cylinder is to hold a gallon: find the radius. T. M. 10 XXIV. SEGMENTS OF A RIGHT CIRCULAR CYLINDER. RING. 254. There are certain segments of a right circular cylinder, the volumes of which can be found by very simple Rules, as we will now shew. 255. Suppose a right circular cylinder cut into two parts by a plane parallel to the axis; then each part has a segment of a circle for base. The volume of each part may be found by the Rule of Art. 246. 256. Suppose a solid has been obtained by cutting a right circular cylinder by a plane, inclined to the axis, which does not meet the base of the cylinder. Let the straight F line CD, drawn from the centre of the base at right angles to the base to meet the other plane, be called the height of the solid. Then the Rule for finding the volume of this solid is the same as that given in Art. 246. Thus we may say that the height of the solid is the portion of the axis of the cylinder which is con- A tained between the two ends. D G B 257. The preceding Rule may be easily justified. For if we suppose a plane drawn through D parallel to the base of the cylinder, it will cut off a wedge-shaped slice, which may be so adjusted to the remaining solid as to form a complete right circular cylinder with the height CD. 258. In the diagram of Art. 256, it is easy to see that CD is half the sum of AF and BG; that is, the height is equal to half the sum of the greatest and the least straight lines which can be drawn on the solid parallel to the axis of the cylinder. See Art. 163. 259. Suppose a solid has been obtained by cutting a right circular cylinder by two planes, inclined to the axis, which do not meet each other. The volume of the solid will be found by multiplying the base of the cylinder by the height of the solid; where by the height of the solid we must understand the portion of the axis of the cylinder which is contained between the two ends. The Rule follows from the fact that the solid may be supposed to be the difference of two solids of the kind considered in Art. 256. 260. Suppose we have a solid like that represented in the diagram of Art. 256, and that it is bent round until A and F meet: we obtain a solid resembling a solid ring; and thus a ring may be described roughly as a cylinder bent round until the ends meet. This is not exact, but it will serve to illustrate the Rule which we shall now give. 261. To find the volume of a solid ring. Multiply the area of a circular section of the ring by the length of the ring. The circular section is sometimes called the cross section. The length of the ring is the length of the circumference of the circle which passes through the centres of all the cross sections; or it may be described as half the sum of the inner and outer boundaries of the ring. See Art. 258. 262. Examples: (1) The radius of the circular section of a ring is one inch, and the length of the ring is ten inches. The area of the circular section of the ring is 31416 square inches; therefore the volume of the ring in cubic inches is 10 × 31416, that is, 31416. |