684. Theorem. The volume of a triangular prism is equal to the product of its base and altitude. V=Bh. Given the triangular prism A'-ACD. To prove V=Bh, where V denotes volume, B area of base, and h altitude. Suggestion. Complete the parallel epiped A'-ACED. Show that prism Apply § 683. A E T E 685. Theorem. The volume of any prism is equal to the product of its base and altitude. V=Bh. Given any prism AD'. To prove V = Bh, where V denotes volume, B area of base, and h altitude. Suggestion. Show that the prism can be divided into triangular prisms by diagonal planes. Add the prisms thus formed. 686. Theorem. Prisms having equivalent. bases and equal altitudes are equivalent. F B E' D' 687. Theorem. Prisms having equivalent bases are to each other as their altitudes. 688. Theorem. Prisms having equal altitudes are to each other as their bases. EXERCISES 1. Find the volume of a prism whose base is a regular hexagon 6 in. on a side, and whose altitude is 24 in. 2. In a prism given V = 226 and B = 43.6; find h. 3. The sides of a right section of a triangular prism are 4 in., 5 in., and 7 in. Find the volume if a lateral edge is 16 in. 4. The cost of digging a ditch, including all expenses and profits, is estimated at 27 cents a cubic yard. Find the cost of digging a ditch 15 mi. long, 10 ft. wide at the bottom, 20 ft. at the top, and 6 ft. deep. The cross section is a trapezoid. Ans. $71,280. 5. Show that an oblique prism of wood may be changed into a right prism by cutting along a right section and interchanging the two parts. 6. One of the concrete pillars to support a floor in a concrete building is 12 ft. high and has as a cross section a regular hexagon 8 in. on a side. Find its weight if the concrete weighs 138 lb. per cubic foot. Ans. 1912 lb. 7. A flow of 300 gallons per second will supply water for a stream of what depth, if the stream is 4 ft. wide and flows 5 miles per hour? 9. One cubic inch of steel weighs 0.29 lb. An I-beam has a cross section as shown in the figure and a length of 22 ft. Find its weight. 10. An iron casting shrinks in. per linear foot in cooling down to 70 degrees Fahrenheit. inches is the shrinkage per cubic foot? How many cubic Ans. 16 in. -20 11. How many cubic yards of soil will it take to fill in a lot 50 ft. by 100 ft. if it is to be raised 3 ft. in the rear and gradually sloped to the front where it is to be 1 ft. deep? Ans. 416 cu. yd. 12. In a regular triangular prism, the edge of the base and the altitude are equal. Find these dimensions if the volume is 128√3 cu. in. 13. In a regular triangular prism, the altitude is 6 in. more than the edge of the base. Find the dimensions if the volume is 224√3 cu. in. 14. Find the volume of a regular hexagonal prism whose base has an area of 37.5 sq. ft., and D whose altitude equals an edge of the base. 15. The perpendicular drawn to the lower base of a truncated triangular prism from the intersection of the medians of the upper base, equals one third the sum of the three lateral edges. A 16. The volume of any oblique prism is equal to the product of the area of a right section by the length of a lateral edge. 17. The volume of a regular hexagonal prism is 30√3 cu. in. and its lateral area is 180 sq. in. Find its altitude and base edge. 18. The volume of a triangular prism is equal to the area of a lateral face times one-half the perpendicular drawn to that face from the opposite edge. 689. Theorem. The volume of a circular cylinder is equal to the product of its base and altitude. Given a circular cylinder. V=Bh. To prove that V=Bh, where V denotes volume, B area of base, and h altitude. B Proof. Inscribe a prism, whose base is a regular polygon, in the cylinder, and let V' denote its volume and B' the area of its base. By doubling indefinitely the number of faces of the prism, V'-V, and B'→B. §§ 651 (3), 489 B'h-Bh. § 485 (2) Then But V' = B'h, being variables that are always equal. § 685 1. Show that the volume of a hollow cylinder with outer diameter D, inner diameter d, and altitude h, is given by the formula, V=Th(D2-d2) =πh(D+d) (D−d). (See Ex. 8, p. 249.) 2. A cylindrical oil tank 3 ft. in diameter and 10 ft. long will contain how many gallons? (1 gal. = 231 cu. in.) Ans. 528.8. 3. The outer diameter of a hollow cast iron shaft is 18 in.; and its inner diameter is 10 in. Calculate its weight if the length is 20 ft. and cast iron weighs 0.26 lb. per cubic inch. 4. A peck measure is to have a diameter of 8 in. How deep should it be? (1 bu. =2150.42 cu. in.) 5. Water is flowing at the rate of 10 miles per hour through a pipe of diameter 16 in. into a rectangular reservoir 197 yd. long and 87 yd. wide. Calculate the time in which the surface of the water in the tank will be raised 3 in. Ans. 31.38 minutes. 6. A certain handbook gives the following "rules of thumb" for finding the volume in gallons of a cylindrical tank: (1) V = (diameter in feet)2×53×(height in feet). (2) V = (diameter in feet)2 (height in inches) less 2% of the product. Find the per cent of error for each rule. 7. Find the height of a 10-gallon wash boiler whose base is 10 in. wide with semicircular ends, the length of the straight part of the sides being 94 in. Ans. 13.5 in. 8. A cylinder to cool lard is 4 ft. in diameter and 9 ft. long and makes four revolutions per minute. At each revolution, the hot lard cools upon the surface to a depth of in. How many pounds of lard will it cool in one hour if 1 cu. ft. of lard weighs 56 pounds? 9. If a tank 5 ft. in diameter and 10 ft. deep holds 10,000 pounds of lard, what will be the depth of a tank of 2000 pounds capacity if its diameter is 3 ft.? If this tank has a jacket around it on the bottom and sides 3 in. from the surface of the tank, how many gallons of water will the space between the jacket and the tank hold? 10. A conduit made of concrete has a cross section with dimensions as shown in the figure. How many cubic yards of concrete are used in making one mile of this conduit? 11. The cylindrical water tower shown in the figure is at Long Beach, N.Y. Its diameter is 34 ft., height 150 ft., and it is said to have a capacity of 1,020,000 gallons. Is the capacity given correct? 12. Does a cylindrical water tank 42 in. in diameter and 14 ft. long hold 1000 gallons? 10 -72" " 10 13. The following "rule of thumb" is used for finding the weight of round iron. The weight of round iron in pounds per foot equals the square of the diameter in quarter inches divided by 6. Find the per cent of error in using the rule if iron weighs 0.28 lb. per cubic inch. 14. How much water will a horizontal steam boiler 5 ft. in diameter and 16 ft. long with 70 tubes of diameter 3 in. running lengthwise, hold if one third of the volume is for steam? 15. Find the height of a cylindrical oil tank with a diameter of 16 in. to hold one barrel. Ans. 36.2 in. 16. A tool steel ring for a steam cylinder is forged from round stock 3 in. in diameter. Find the length of -12 17. The segment in the figure is a counter-balance 51⁄2 in. thick. Find its weight if made of cast-iron weighing 0.26 lb. per cubic inch. Ans. 228.5 lb. A 18. Find the weight of a hollow hexagonal bar 16 ft. long and weighing 0.28 lb. per cubic inch. The cross section is a regular hexagon 1 in. on a side, with a circle 1 in. in diameter at the center. Ans. 152 lb. n B 60° 84" SIMILAR PRISMS AND CYLINDERS 690. Definitions. Two right prisms having similar polygons for bases, and whose altitudes are in the same ratio as two corresponding base edges, are similar. Two right circular cylinders are similar if their altitudes are in the same ratio as the radii of their bases. 691. Theorem. The volumes of two similar right prisms are in the same ratio as the cubes of corresponding base edges. Given two similar right prisms P and P', having altitudes h and h', bases B and B', and two corresponding base edges e and e' |