653. Theorem. The lateral area of a right circular cylinder is equal to the product of its altitude and the circumference of its base. That is, S=ch, where S denotes lateral area, c circumference of base, and h altitude. EXERCISES 1. Show that the lateral area and the total area of a right circular cylinder of altitude h and radius r, are given by the formulas: S=2πrh, T=2πrh+2πr2=2πr(h+r). In the following use the notation of Exercise 1. 2. Given r=8 and h=10; find S and T. 8. Solve the formula T=2πrh+2πr2, (1) for r in terms of h and T, (2) for h in terms of r and T. 9. From the formulas of Exercise 1, find r in terms of S and T. -24" 10. A peck measure made of sheet iron has a diameter of 8 in. and a depth of 10.7 in. Find the number of square inches of sheet iron in it. 11. Find the area (no cover) of a wash boiler if the bottom is in the form of a rectangle with a semicircle at each end. The rectangle is 10 in. by 14 in., and the depth of the boiler is 16 in. 12. A steam boiler has a diameter of 72 in., is 18 ft. long. and contains 70 tubes each having a diameter of 4 in. extending lengthwise of the boiler. Find the heating surface of the boiler, using "one-half" in the rule below. Ans. 1505 sq. ft. −1·4 " RULE. In finding the heating surface of a horizontal boiler, it is customary to take one-half to two-thirds the lateral area of the shell, the lateral area of the tubes, one-half to two-thirds the area of the ends of the boiler, and subtract the area of both ends of the tubes. 13. Find the heating surface of a horizontal tubular boiler 5 ft. in diameter, and having 52 tubes 2 in. in diameter. thirds" in the preceding rule. 12 ft. long, Use "twoAns. 556.7 sq. ft. 14. The base of a right prism is a triangle whose sides are 12 ft., 15 ft., and 17 ft., and its altitude is 8 ft. Find its lateral area and total area. 15. Find the lateral area of a regular hexagonal prism each side of whose base is 4 in. and whose altitude is 16 in. Find the total area. 16. Derive the formula S=0.2618dl, where S is the area of the surface of a cylindrical pipe in square feet, d the diameter in inches, and I the length in feet. 17. Six lines of steam pipes of diameter 2 in. extend along one side and an end of a room 40 ft. by 30 ft. Find the number of square feet of heating surface. Blank Box 18. Small metal boxes used for various purposes are made from sheet metal as follows: Blanks of the proper shape are first cut from the sheet metal. These are then pressed into the required form in a die. In computing the size of the blank, it is assumed that it has the same area as the finished article. Find the diameter of the blank for the cover of a tin pail for holding lard, if the cover has a diameter of 61⁄2 in. and the flange is § in. 19. A cylindrical metal box for holding paper fasteners is 12 in. in diameter and 11⁄2 in. deep, and the cover has a 1⁄2-inch flange. Find the diameter of the blanks for box and cover. Ans. 3.68 in.; 2.56 in. 20. What is the locus of points 12 in. distant from two parallel lines 18 in. apart? 21. Holes for rivets are often punched in metal plates. The pressure required for such work, when the material is wrought iron, is 55,000 pounds per square inch of the cut surface in. For example, a hole having a circumference of 2 in. punched in a 1-in. plate would require a pressure of 2××55,000 lb., that is, the area of the cylindrical surface sheared off times 55,000 lb. Find the pressure required to punch a 1-in. round hole through a piece of sheet iron-in. thick. 22. Find the pressure necessary to punch a 2-inch round hole through a boiler plate in. thick if 60,000 pounds pressure is required per square inch of surface cut. 23. A rectangular box is to be made of sheet steel in. thick. Find the number of pounds the punch press must strike in order to cut out the blank if it is rectangular 5 in. by 24 in., has squares in. on a side cut from each corner, and has two holes in. in diameter punched in the bottom. Use 60,000 pounds per square inch, Ans. 30,297 lb. CONGRUENT AND EQUIVALENT SOLIDS 654. Congruent Solids. Two solids are said to be congruent if they can be made to coincide completely in all their parts. 655. Equivalent solids are those having the same volume. 656. Theorem. Two prisms are congruent if three faces including a triedral angle of one are congruent respectively to three faces including a triedral angle of the other, and are arranged in the same order. Given prism AI and A'I', with face AJ face A'J', face AG face A'G', and face AD face A'D'. Place the prism A'I' upon prism AI so that triedral ZA' shall coincide with its congruent triedral ZA. Then face A'J' coincides with its congruent face AJ; A'G' with its congruent face AG; and A'D' with its congruent face AD. And points F', G', and J' fall upon F, G, and J respectively. Further C'H' coincides with CH. Hence H' falls upon H. Why? Similarly I' falls upon I. ..prism AI prism A'I'. § 654 657. Theorem. Two right prisms, or two right cylinders, are congruent if they have congruent bases and equal altitudes. 658. Truncated Prism. A portion of a prism included between the base and a section oblique to the base is called a truncated prism. 659. Theorem. Two truncated prisms are congruent if three faces including a triedral angle of one are congruent respectively to the three faces including a triedral angle of the other, and are arranged in the same order. 660. Theorem. An oblique prism is equivalent to a right prism whose base is a right section of the oblique prism, and whose altitude is a lateral edge of the oblique prism. Given the oblique prism AD'; also the right prism GJ' whose base GJ is a right section of the prism AD', and whose altitude is equal to a lateral edge AA' of prism AD'. זי G K H J Α' E D' G LB' IC Likewise show face BI face B'I', and face AD face A'D'. Then prism AJ prism A'J'. Prism AJ+prism GD' = prism A'J'+prism GD'. .. prism AD' = prism GJ'. EXERCISES 1. Compare the theorem of § 660 with the theorem of § 354. § 656 Why? Why? 2. If a wooden beam has a rectangular right cross section, show that if it is sawed lengthwise along a diagonal plane the two prisms formed are congruent. 3. In a truncated prism having a parallelogram for base, the sum of two opposite lateral edges is equal to the sum of a the other two opposite lateral edges. Prove a+c=2e, and b+d=2e. b PARALLELEPIPEDS 661. A parallelepiped is a prism whose bases are parallelograms. 662. A right parallelepiped is a parallelepiped whose lateral edges are perpendicular to its bases. 663. An oblique parallelepiped is a parallelepiped whose lateral edges are oblique to its bases. 664. A rectangular parallelepiped is a right parallelepiped whose bases are rectangles. 665. A cube is a rectangular parallelepiped whose edges are all equal. OBLIQUE PARALLELEPIPED CUBE RECTANGULAR 666. The dimensions of a rectangular parallelepiped are the lengths of the three edges drawn from one vertex. These dimensions are often called length, breadth, and height. 667. A diagonal of a parallelepiped is the line from any vertex to a vertex not in the same face. 668. The volume of any solid is the numerical measure of its magnitude in terms of some unit of measure. The unit of measure usually taken is a cube one linear unit on an edge. 669. Ratio between Solids. By the ratio of one solid to another is meant the ratio of their numerical measures. 670. Prove the following facts about parallelepipeds: (1) All the faces (including the bases) of a parallelepiped are parallelograms. (2) A parallelepiped is bounded by three pairs of parallel congruent parallelograms. |
