5. The base edges of a triangular pyramid are 6 in., 8 in., and 10 in. Find its volume if the altitude is 15 in. 6. Find the volume of a circular cone whose radius is 5 in. and altitude 10 in. 7. Find the volume of a circular cone whose altitude is 70 ft. and the circumference of whose base is 31 ft. 8. Derive a formula for finding the radius of a circular cone in terms of the volume and altitude. 9. Derive a formula for finding the altitude of a circular cone in terms of the volume and radius. 10. A right circular cone has a slant height s and a radius r; find its volume V. Ans. V=}πr2√s2 —r2. 11. The Pyramid of Cheops has a square base 720 ft. on a side, and an altitude of 480 ft. Find the number of cubic yards in it. 12. What is the locus of the vertices of all pyramids having the same base and equal volumes? 13. Find the total area and the volume of the solid generated by revolving an equilateral traingle of side a about one side. Find the values if a = 2 in. Ans. Area = a2π √3; 21.765 sq. in. Volume = {a3T; 6.283 cu. in. 14. Find the total area and the volume of the solid generated by revolving a parallelogram with sides 22 in. and 16 in., and larger angle 120°, about one of the longer sides. Ans. 3308.3- sq. in.; 13270+ cu. in. How far from 15. A pyramid whose base is a square 4 ft. on a side, and whose altitude is 12 ft. is bisected by a plane parallel to the base. the vertex is the plane? 16. A circular sheet of copper 3 ft. in diameter is cut in half, and each half formed into a cone. These are placed base to base as shown in the figure. This is to be used as a float. Find the number of pounds it will support if the sheet copper weighs 0.92 lb. per square foot and water weighs 62.5 lb. per cubic foot. 17. Hard coal dumped in a pile lies at an angle of 30° with the horizontal. Estimate the number of tons in a pile in the shape of a right circular cone having an altitude of 10 ft. Large egg size weighs 38 lb. per cubic foot. Ans. About 60 tons. 18. Show how to construct a triangular pyramid equivalent to any given pyramid having a polygon as base 741. Theorem. The volume V of a frustum of a pyramid, or circular cone, of bases B1 and B2 and altitude h, is given by the formula V =}h(B1+B2+√/B¿B2). Given the frustum of a pyramid, or of a circular cone. To prove V=h(B1+B2+√B ̧B1⁄2), where V denotes volume, h altitude, and B1 and B2 the areas of the lower base and upper base, respectively. Proof. Complete the pyramid, or cone, and let m denote the altitude of the portion above the frustum in each case, and V2 the volume of that portion. Also let V1 denote the volume of the entire pyramid, or cone. 1 It remains to eliminate m. To do this one more equation is needed. This is given by 742. Theorem. The volume V of a frustum of a circular cone having bases with diameters d, and d2, and radii r1 and r2, and an altitude h, is given by the formulas: (1) V=}πh(rı2+r22+rır2), (2) V=2h(d2+d22+d1d2). Substitute Tri2 and πr1⁄22 for B1 and B2 respectively of § 741 to find (1). Use πd and πd22 for (2). EXERCISES 1. Find the volume of a frustum of a pyramid with square bases having edges of 9 ft. and 4 ft. respectively, and altitude 12 ft. 2. The radii of the bases of a frustum of a right circular cone are respectively 10 cm. and 12 cm. Find its volume if the altitude is 16 cm. Find its lateral area. 3. Find the volume of a frustum of a regular hexagonal pyramid the bases being respectively 8 in. and 6 in. on an edge, and the altitude 10 in. 4. The frustum of a cone has an altitude of 8 in. and the radii of its bases are 7 in. and 4 in. respectively. Find its volume. Does the frustum have to be cut from a right cone in order that its volume can be found? 5. Derive a formula for finding the altitude of a frustum of a circular cone in terms of the volume and the radii of the bases. 6. The diameter of the top of a water pail is 12 in., the diameter of the bottom 10 in., and the altitude 10 in. How many quarts will the pail hold? One quart is 57.75 cu. in. Ans. 17.33-. 7. Find the weight of an iron casting in the form of a right circular cone of diameter 8 in. and altitude 12 in., with a circular hole having a diameter of 2 in. through the center. (Cast iron weighs 0.26 lb. per cubic inch.) Ans. 44.1 lb. 12' 8. A tank is made in the form of an inverted frustum of a cone. The slant height is 14 ft. and makes an angle of 60° with the horizontal, and the lower base is a circle 20 ft. in diameter. Find the volume of the tank in barrels. Use 1 bbl. 4.211 cu. ft. Ans. 1685.4 bbl. 9. A tank is in the form of an inverted cone with its vertex angle 60° and its axis vertical. Find the depth of water in the cone if it has been running in for 10 minutes at the rate of 8 cu. ft. per minute. SIMILAR PYRAMIDS AND CONES 743. Similar Pyramids. Pyramids that are formed from the same pyramidal surface by parallel planes are similar. 744. Similar Cones. Cones that are formed from the same conical surface by parallel planes are similar. 745. Theorem. The lateral areas, or the total areas, of two similar regular pyramids, or right circular cones, are to each other as the squares of two corresponding lines, where the corresponding lines may be edges, elements, altitudes, slant heights or radii. One case only will be proved. The other cases are left as exercises. Given two similar regular pyramids O-ACDEF and O-A'C'D'E'F'. D' F M A C S' denote the lateral areas of O-ACDEF and O-A'C'D'E'F' respectively, h and h' altitudes, and T and T' total areas. Proof. If p and p' are the perimeters of the respective bases, and s and s' the slant heights, S=2ps, and S'p's'. §711 Further, if B and B' are the areas of the respective bases, and Why? s' h' 746. Theorem. The volumes of two similar pyramids, or circular cones, are to each other as the cubes of two corresponding lines; where the corresponding lines may be edges, elements, altitudes, slant heights, or radii. Using V and V' for the respective volumes, B and B' for bases, and h and h' for altitudes, 747. Definition. V Bh B = V' B'h' B' h' h2 V h3 Why? h'2 V'h'3 Why? The vertex angle of a right circular cone is twice the angle between an element and the axis. 748. Theorem. All right circular cones with equal verter angles are similar. EXERCISES 1. Two cones generated by similar right triangles revolving about corresponding sides are similar. 2. Two cones are generated by revolving two similar right triangles about corresponding sides of length 7 in. and 10 in. respectively. Find the ratio of the lateral areas of the cones. The ratio of the total areas. The ratio of the volumes. 3. The total area of a right circular cone is 300 sq. in. and its altitude is 8 in. Find the total area of the cone cut off by a plane parallel to the base and 6 in. from the vertex. 4. How far from the vertex of a right circular cone, or a regular pyramid, of altitude h must a plane be passed parallel to the base so that the lateral area of the part cut off shall be one-half that of the original? 5. From a given pyramid, cut off a frustum whose volume shall be that of the given pyramid. Must the pyramid be regular? 6. The volumes of two pyramids are 343 cu. in. and 1000 cu. in. respectively. The lateral area of the smaller is 100 sq. in. Find the lateral area of the larger. 7. The lateral areas of two similar right circular cones are in the ratio of 964. Find the volume of the smaller if the volume of the larger is 1024 cu. in. |