Proposition 24. Volume of a Frustum 162. Theorem. The volume of a frustum of a circular cone is one third the product of the altitude of the frustum by the sum of the areas of the bases and the mean proportional between them. Given a frustum of a circular cone; V, the volume; B, B', the areas of the bases; and h, the altitude. Prove that V = {h(B+B'+√BB'). Proof. Let C be the volume of the cone from which the frustum is cut; and let C' and h' be the volume and altitude respectively of the small cone remaining after the frustum is removed. VB-VB' = B-B' Then V V = } | Bh+(B− B')h√B'(√B+√B') B-B' Simplifying, V= {h(B+B'+√BB'). Ax. 5 Since Br2 and B'r' 2, the above formula may be written V= πh (r2+r22 + rr). This subsidiary proposition may be omitted, if desired. Exercises. Numerical Computations Find the lateral areas of cones of revolution, given the slant heights and the circumferences of the bases respectively as follows: 1. 4 in., 5 in. 2. 4.7 in., 6.2 in. 3. 3 ft. 6 in., 5 ft. Find the lateral areas of cones of revolution, given the slant heights and the radii of the bases respectively as follows: 4. 2.5 in., 2.1 in. 5. 5.8 in., 5.6 in. 6. 3 ft. 3 in., 7 in. Find the total areas of cones of revolution, given the slant heights and the radii of the bases respectively as follows: Find the volumes of circular cones, given the altitudes and the areas of the bases respectively as follows: 13. 4 in., 8 sq. in. 14. 6 in., 10 sq. in. 15. 6 in., 5.8 sq. in. 16. 8.1 in., 18.8 sq. in. Find the volumes of circular cones, given the altitudes and the radii of the bases respectively as follows: 23. How many cubic feet are there in a conical tent which is 14 ft. in diameter and 9 ft. high? 24. How many cubic feet are there in a conical pile of earth which is 18 ft. in diameter and 10 ft. high? 25. An isosceles triangle whose altitude is 4 in. and whose equal sides are each 5 in. revolves about the base. Find the volume of the double cone thus formed. Exercises. Formulas Deduce, if possible, formulas for the following, stating the impossible cases, if any: 1. The lateral area of a cone of revolution in terms of the radius of the base and the altitude. 2. The slant height of a cone of revolution in terms of the lateral area and the circumference of the base. 3. The slant height of a cone of revolution in terms of the lateral area and the radius of the base. 4. The radius of the base of a cone of revolution in terms of the lateral area and the slant height. 5. The slant height of a cone of revolution in terms of the total area and the radius of the base. 6. The circumference of the base of a circular cylinder in terms of the lateral area and the slant height. 7. The volume of a cylinder in terms of the altitude and the lateral area. 8. The altitude of a circular cone in terms of the volume and the area of the base. 9. The area of the base of a circular cone in terms of the volume and the altitude. 10. The altitude of a circular cylinder in terms of the volume and the radius of the base. 11. The radius of the base of a circular cylinder in terms of the volume and the altitude. 12. The volume of a cone of revolution in terms of the slant height and the radius of the base. 13. The slant height and the altitude of a cone of revolution in terms of the volume and the circumference of the base. Exercises. Theory of the Cone 1. Every section of a cone made by a plane passing through the vertex is a triangle. 2. The axis of a circular cone passes through the center of every section which is parallel to the base. 3. Defining similar cones of revolution as cones formed by the revolution of similar right triangles about corresponding sides, prove that the lateral areas of two similar cones of revolution are to each other as the squares of their altitudes. 4. In Ex. 3, consider the case of the total areas. 5. Consider Ex. 3 with "slant heights" substituted for "'altitudes." 6. Consider Ex. 3 with "radii" substituted for "altitudes." 7. Consider Ex. 3 with respect to volumes instead of lateral areas, changing the statement as may be necessary. 8. If the lateral surface of a cone of revolution is cut along one of the elements and unrolled on a plane, show that it becomes a sector of a circle. Show also that there can be deduced a formula for the area of a sector of a circle that shall be the same as the formula for the lateral area of a cone. 9. Deduce a formula for finding the altitude of a fristum of a circular cone in terms of the volume and the areas of the bases. If the subsidiary proposition of § 162 was not taken, omit Exs. 9 and 10. 10. Deduce a formula for finding the altitude of a frustum of a cone of revolution in terms of the volume and the radii of the bases. Exercises. Industrial Problems 1. A steamer's funnel 4 ft. 8 in. in diameter is built up of four equal plates in girth, with a lap at each joint of 1 in. Find one dimension of each plate. 2. There is a rule for calculating the strongest beam that can be cut from a cylindric log, as follows: Trisect the diameter AB, and at the points of division P, Q erect the Is PC, QD on opposite sides A P of AB, and meeting the circle in C and D. Then ADBC is a section of the strongest beam. Given that the diameter AB of a log is 18 in., calculate the dimensions AD and DB of the strongest beam that can be cut from the log. 3. A tubular boiler has 128 tubes each 33 in. in diameter and 16 ft. long. Find the area of the total surface of the tubes to the nearest square foot. 4. In a room of a factory heated by steam pipes, there are 280 ft. of 2-inch pipe, 36 ft. of 3-inch pipe, and 6 ft. of 44-inch feed pipe. Find the area of the total heating surface to the nearest square foot. 5. A triangular plate of wrought iron in. thick is 3 ft. 4 in. on each side. If the weight of a plate 1 ft. square and in. thick is 5 lb., what is the weight, to the nearest pound, of the triangular plate? 6. A cylinder 18 in. in diameter is required to hold 60 gal. of water. Allowing 231 cu. in. to the gallon, what must be the height of the cylinder to the nearest 0.1 in.? 7. The water level of an upright cylindric boiler is 1 ft. 4 in. below the top of the boiler. If the cross-section area of the boiler is 14 sq. ft., what is the volume of the steam space above the water? |