pyramid is 20 ft., the length of each side of its square base is 40 ft., and of each side of its top is 16 ft.; find its volume. (13) The slant height of the spire on a church tower, in the shape of the frustum of an hexagonal pyramid, is 20 ft.; the length of each side of its base is 5 ft., and of its top 2 ft. How many square feet of lead will be required for covering its side faces and top? (14) Find the expense of polishing the curved surface of a column of marble, in the shape of the frustum of a cone, whose slant height is 12 ft., and the radii of the circular ends are 3 ft. 6 in. and 2 ft. 4 in. respectively, at 2s. 6d. per sq. ft. (15) Find the cost of the mast of a ship, in the shape of the frustum of a cone, at 2s. 6d. per cub. ft., when its perpendicular height is 51 ft., and the circumference at one end is 5 ft. 6 in., and at the other end 1 ft. 10 in. (16) A bucket is 1 ft. 4 in. deep; its diameter at the top is 1 ft. 6 in., and at the bottom 1 ft. Find how many gallons of water it will hold, when there are 6 gallons to a cub. ft. (17) How many cubic inches of lead will be required for covering the curved surface and the bottom of a vessel, in the shape of the frustum of a cone, with lead of an in. thick; the diameter of the top of the vessel is 7 ft., and of the bottom 3 ft. 6 in., and the perpendicular depth is 2 ft. 4 in. ? (18) Supposing that an ordinary glass tumbler, in the shape of the frustum of a cone, is 31⁄2 in. wide at the top, and 2 in. wide at the bottom, and that its depth is 4 in.; find how many cubic inches of water it will hold. (19) A marble column, in the shape of the frustum of a square pyramid, is 18 ft. high; the length of each side of its base is 4 ft., and of its top 24 ft. Find its volume. (20) An iron vessel without lid, in the shape of the frustum of a cone, has the following internal dimensions :— The diameter at the top is 1 ft. 9 in., at the bottom 7 in., and the depth 1 ft. 8 in.; whilst its corresponding external dimensions are 2 ft., 9 in., and 1 ft. 10 in. Find how many cubic inches of iron were used in its construction. (21*) Find the content of a cask, in the form of a conical frustum, the radii of its ends being 2 ft. and 3 ft., and the height 5 ft. XXIII. THE WEDGE. Definitions.-A wedge is a solid of which the base or thick end ABCD is a rectangle; the two ends AEC, BFD are triangles; and the two opposite D faces AEFB, CEFD meet in the edge or thin end EF, and are either rectangles or trapezoids. In the common wedge we have EF=AB=CD, and the solid becomes a triangular prism, of which its ends AEC, BDF are equal and parallel isosceles triangles, B F and its base and two side faces are rectangles. C E A The volume of a common wedge is the same as that of a triangular prism, and may therefore be found by multiplying the area of its end BFD by EF, the length of the wedge. But when EF is not equal to AB or CD, then the two side faces EABF, ECDF are trapezoids; and the volume of such a wedge is found by the rule given below. The perpendicular height—or, as it is often called simply, the height is the perpendicular drawn from any point in the edge EF to the base ABCD. [In some cases the base is a trapezoid; but it is desirable that the Student should confine his attention at present to the consideration of the simpler kinds of the wedge.] RULES. (1) To find the volume of a wedge. To twice the length of the base or thick end, add the length of the thin end or edge; multiply the sum by the breadth of the base or thick end, and the product by the height of the wedge; one-sixth of the result is the volume of the wedge. (2) To find the whole surface of a wedge. Find the area of each of its sides or parts, and add them together. Note. To find the volume of a wedge, when the area of a section of it, perpendicular to the edge, and also the length of its base and edge, are given.-Add together the edge and twice the length of the base, and divide the sum by 3, and the quotient is the mean length of the wedge. Multiply the area of the section by the mean length, and the product is the volume of the wedge. Example 1.-Find the volume of a wedge, when the length and breadth of its base are 26 in. and 18 in., the length of the thin end or edge is 15 in., and the height is 28 in. Now, 26 x 2+15=52+15=67 in. Then, volume of wedge=67 × 18 × 28 × 1=5628 cubic in. =3 cub. ft. 444 cub. in. Example 2.-The edge of a wedge is 110 in.; the length of the base is 70 in., and its breadth is 30 in.; the height of the wedge is 24.8 in. Find its volume. Now, 2 x 70+110=140+110=250 in. Then, volume of wedge=250 × 30 × 24·8 × 1=31000 cub. in.=17 cub. ft. 1624 cub. in. Formulæ I. Volume (2AB+EFX AC x height x ). II. Whole surface=sum of the areas of all its sides or parts. [N.B.-When the word height occurs alone, it refers to the perpendicular height-that is, to the straight line drawn from the edge perpendicular to the base.] EXAMPLES. (1) Find the volume of a wedge, of which the base is a rectangle, whose length is 12 in. and breadth 6 in.; the edge is 12 in., and the height of the wedge is 10 in. (2) The edge of a wedge is 1 ft. 8 in.; the base of the wedge is a square, each side of which measures 1 ft. 8 in.; the height of the wedge is 3 ft. Find its volume. (3) Find the volume of a wedge, whose base is 3 ft. 6 in. long and 1 ft. 8 in. broad; the length of the edge is 2 ft., and the height of the wedge is 2 ft. 6 in. (4) The length of the rectangular base of a wedge is 20 in., and its breadth is 10 in.; the edge is 11 in., and the height of the wedge is 2 ft. Find its volume. (5) The edge of a wedge is 1 ft. 3 in.; the length of the base is 2 ft.; the area of a section of the wedge made by a plane perpendicular to the edge is 1 sq. ft. 36 sq. in. Find the volume of the wedge. (6) The volume of a wedge is 1020 cub. in.; its edge is 15 in.; its base measures 18 in. in length and 12 in. in breadth. Find its height. (7) Find the volume of a wedge, when the following dimensions are given : AB=4 ft. 6 in.; EF=3 ft.; AC=1 ft. 4 in.; and AE= 1 ft. 5 in. (8) Find the volume of a wedge, when the following dimensions are given : AB=6.5 ft.; EF=8 ft.; AC=4.25 ft.; and the line drawn from E to the middle of Ac=12 ft. (9**) The length of the edge of a wedge is 5 in.; the length of the base is 3 in., and its breadth is 2 in.; the height of the wedge is 4 in. Find its volume. (10*) Find the solid contents of a wedge whose edge and length of the base are each 18 in.; the breadth of the base is 8 in., and the height 30 in. (11) Find the whole surface of a wedge, when its base is 5 ft. 6 in. long and 2 ft. broad; the edge is 5 ft. 6 in., and the height of the wedge is 2 ft. 11 in. XXIV. THE PRISMOID, OR FRUSTUM OF A WEDGE. Definitions. A prismoid, or frustum of a wedge, is the solid that remains after a smaller wedge has been cut off by a plane parallel to the also called the base; then its opposite face HEFG is called the top. Both these side faces are sometimes called the ends of a prismoid. The perpendicular height-or, as it is generally called simply, the height-is the perpendicular drawn from one end HF to the other end BD. If LKNM is the middle section of the prismoid—that is, a section parallel to and equally distant from each end-then its length KN or ML= ; and its breadth LK or MN AD+ER 2 AB+EF |