(49**) How many yards of canvas, yd. wide, will be required to make a conical tent 8 ft. high, and 10 ft. in diameter? How many cubic feet of air will the same contain ? (50*) A cylindrical stick, in. thick, is sharpened at one end into the shape of a cone, the slant side of which is É of an in. long; how much wood is cut away in doing this? 5 8 (51) The radius of a cylinder is 7 ft., and its height is 12 ft., find the perpendicular height of a cone that shall have the same base and area of curved surface as the cylinder. (52*) A square tower, 21 ft. on each side, is to have either a flat roof covered with sheet-lead which costs 6d. per sq. ft., or a pyramidal roof, whose vertical height is 10 ft., covered with slates which cost 18s. 9d. per hundred, and each of which has an exposed surface of 10 in. by 9 in.; find the cost in each case. XXII. THE FRUSTUM OF A RIGHT-CONE OR RIGHT-PYRAMID. Definitions. The frustum of a cone or pyramid is that part of the cone or pyramid which remains when its top part has been cut off by a plane parallel to the base. The top part cut off, in the one case, is a cone, and in the other a pyramid. The base AB and its opposite face DE are called the ends of the frustum. GF, the perpendicular drawn from one end to the other, is called the perpendicular height; or, more generally, simply the height of the frustum. AD is, in the case of the cone (fig. 1), the slant height, and is the hypothenuse of a right-angled triangle DLA, of which the perpendicular is DL, the height of the frustum; and the base is AL, the difference between the radii of the two ends. The slant height of the frustum of a pyramid (fig. 2) is measured from the middle of a side of one end to the middle of the corresponding side of the other end. The whole surface of the frustum of a cone (fig. 1) is the areas of the two circular ends added to the area of the curved surface. The whole surface of the frustum of a pyramid (fig. 2) is the areas of the two ends added to the area of the side faces. [N.B. The attention of the Student in this chapter is entirely confined to the consideration of the frustum of a right-cone and of a right-pyramid.] RULES.-(1) To find the volume of the frustum of a cone or pyramid. To the areas of the two ends of the frustum add the square root of their product; multiply the sum by the height of the frustum; and one-third of the product will be the volume. (2) To find the volume of the frustum of a cone, when the diameter or radii of the two ends are given. (a) Add the squares of the radii of the ends to the product of the radii; multiply the sum by the height, and this product by 22; one-third of the result is the volume. (b) Or, add the squares of the diameters of the ends to the product of the diameters; multiply the sum by the height, and this product by 11; onethird of the result is the volume. (3) To find the curved surface of the frustum of a cone, or the area of the side faces of the frustum of a pyramid. Add together the circumferences or perimeters of the two ends; multiply the sum by the slant height of the frustum, and half the product is the area. (4) To find the whole surface of the frustum of a cone or pyramid. To the area of the curved surface or side faces, add the areas of the two ends. Note 7.-For an easy practical rule for finding the volume of the frustum of a cone or pyramid, see Note 2, Prob. 24. Example 1.-Find the volume of the frustum of a cone, when the diameters of its ends are, respectively, 16 ft. and 9 ft., and the height of the frustum is 24 ft. By Rule 1: Area of one end=162×14=256×11; the area of the other end=92×14=81×11; the square root of the product of these areas= √256 × 14 × 81 × 14 =√256 x 81 x 16 x 9=x 144. Add these results, and we have multiplied by the sum of 256, 81, and 144; that is, x 481. Then, volume=1×24 × 481=11x21164-30232 cub. ft. Example 2.-Find the volume of the frustum of a square pyramid, each side of one of its ends being 8 ft., and each side of the other end 4 ft., and the perpendicular height is 10 ft. The area of one end=82=64; the area of the other end =4216; and the square root of the product of these areas=V64x16=8×4–32. Adding together 64, 16, and 32, we have 112. Example 3.-Find the curved surface of the frustum of a cone whose slant height is 16.5 ft., and the circumference of its two ends are, respectively, 14.2 ft. and 11.8 ft. The sum of the circumferences of two ends=14.2 +11.8 =26 ft. Then, curved surface=26 × 16·5×=214.5 sq. ft. Formulæ I. Volume = {area of one end+area of other end + (area of one end x area II. Volume={radius of one end2+radius of other end2+ product of two radii} 22 height X X 7 3 III. Volume = {diameter of one end2 +diameter of other end2+product IV. Curved surface = {circumference of one end + circumference of other end} x slant height 2 V.Whole surface-curved surface + areas of two ends. [Observe that the following questions refer only to the frustum of a right-cone or to that of a right-pyramid, and that it is necessary to know the perpendicular height before finding the volume, and the slant height before finding the curved surface. When the word height occurs alone, it refers to the perpendicular height.] EXAMPLES. Find the volume of the frustum of a cone, when the dimensions given are— (1) Radii of circular ends 5 ft. and 4 ft., and height 6 ft. (2) Radii of circular ends 8 ft. and 6 ft., and height 15 ft. (3) Radii of circular ends 10 ft. 6 in. and 7 ft., and height 12 ft. (4) Radii of circular ends 3 ft. 10 in. and 3 ft., and slant height 2 ft. 2 in. (5) Radii of circular ends 6 ft. 5 in. and 3 ft. 6 in., and slant height 7 ft. 7 in. Find the curved surface of the frustum of a cone, when its dimensions are, respectively— (6) Radii of circular ends 6 ft. and 5 ft., and slant height 14 ft. (7) Radii of circular ends 14 ft. and 7 ft., and slant height 30 ft. (8) Radii of circular ends 14 ft. and 6 ft., and height 15 ft. (9) Radii of circular ends 14 ft. and 7 ft., and height 24 ft. (10) Find the volume of the frustum of a square pyramid whose height is 24 ft., and the sides of its square ends are, respectively, 9 ft. and 4 ft. (11) The length of each side of the base of the frustum of a square pyramid is 16 ft., and of each side of the top is 9 ft.; the height of the frustum is 30 ft. Find its volume. (12) The slant height of the frustum of a square |