692. Theorem. The volumes of two similar right circular cylinders are in the same ratio as the cubes of their altitudes, radii, or diameters. BL-r. C'in' Given two similar right circular cylinders C and C', having volumes V and V', bases B and B', altitudes h and h', radii r and r', and diameters d and d'. 1. Using the notation of § 691 for similar prisms, also S and S' for lateral areas, and T and T'' for total areas, prove the following: 2. Using the notation of § 692 for similar cylinders, also S, S', T, and T'as in Exercise 1, prove the following: 3. Find the ratios of the volumes, the lateral areas, and the total areas of two similar right prisms having hexagonal bases two of whose corresponding edges are 2 in. and 5 in. respectively. 4. Two similar right circular cylinders have radii of 3 in. and 7 in. respectively. Find the ratio of their volumes. Of their lateral areas. Of their total areas. 5. Two similar right circular cylinders have bases whose areas are 20.25 sq. in. and 100 sq. in. respectively. Find the ratio of their volumes. 6. The total areas of two similar right circular cylinders are 625 sq. in. and 324 sq. in. respectively. Find the diameter of the second if the diameter of the first is 6 in. 7. A rectangle having dimensions of 10 in. and 15 in. is revolved, first about the side that is 10 in., and second, about the side that is 15 in. Find the ratio of the volumes of the two cylinders formed. 8. The number of feet of lumber in a log is often based upon a standard log usually 12 ft. long and 24 in. in diameter inside the bark at the small end. If v, d, and I are the volume, diameter, and length respectively of the standard log; and V, D, and L, the corresponding measurements of the log to be measured, then V = vD2L Derive this formula. QUESTIONS 1. What is a prismatic surface? A cylindrical surface? Is the directrix necessarily closed? 2. In § 617, suppose the directrix is coplanar with the generatrix, what is the form of the surface generated? 3. What are the formulas or rules for finding areas of prisms and cylinders? What are the formulas or rules for finding the volumes of each of these solids? 4. Why is there such a close relation between theorems concerning prisms and cylinders? 5. Can you find the area of an oblique circular cylinder? Of an oblique prism? Can you find the volume of each of these? 6. Trace the steps in finding the volume of a parallelepiped. 7. Trace the steps in finding the volume of any prism. Of a cylinder. 8. What effect does it have upon the volume of a prism or cylinder if the base is doubled? If the altitude is doubled? If both base and altitude are doubled? 9. What effect does it have upon the lateral area of a right circular cylinder if the circumference of the base is doubled? If the area of the base is doubled? 10. Which do you consider the more common form in nature, the cylinder or the prism? Name some examples of each form. 11. Which is the more common of these forms in buildings and architecture? In machinery? Give illustrations of triangular prisms, of quadrangular prisms, of pentagonal, of hexagonal, etc. GENERAL EXERCISES COMPUTATIONS 1. Find the number of barrels each of the following cylindrical tanks will hold: (1) diameter 5 ft. and depth 5 ft., (2) diameter 20 ft. and depth 19 ft. (1 bbl. = 3.211 cu. ft.) 2. Find the cost of laying the stonework, at $1.75 per cubic yard, in two abutments for a bridge, each abutment to be 8 ft. high, 3 ft. thick, 20 ft. long at the bottom, and 15 ft. at the top. 3. Find the cost of common brick in the pier with a cross section as shown in the figure, and a height of 12 ft. 6 in., at $7.00 per thousand. cu. ft. -25 Count 20 brick to 1 Top of 4. A stream flowing 5 miles per hour must be how large in cross section to supply water to a depth of 1 in. per week, for 160 acres of land? 5. A water tank in a Pullman car has a vertical section as shown in the figure, and a length of 52 in. capacity in gallons. Ans. 9/16 21 Ans. 28 sq. in. Find its -734" 12" -13% " 68.3 gal. Consider the arc as a part of a cir-↓ cle and apply formula (1) of § 509. 6. The flanges at the joining of two ends of flanged steam pipes 10 in. in inside diameter are bolted together by 14 bolts in. in diameter. If the pressure in the pipes is 200 pounds per square inch, find what each bolt must hold. How much is this per square inch cross section of the bolts? Suppose that the bolts have 10 pitch U.S. standard thread. This makes the root diameter 0.620 in. Ans. 1122 lb.; 3718- lb. 7. Find the diameter of a cylindrical oil tank 40.5 in. high that is to hold 1 barrel. Ans. 15 in. 8. Representing the dimensions of a rectangular solid by x, y, and z, find their values if when each is increased 2 in. the volume is increased 150 cu. in., the face of dimensions x and y is increased 18 sq. in., and the total surface is increased 110 sq. in. Ans. 3 in., 4 in., and 5 in. 9. Two cubes whose edges differ by 1 in. have volumes that differ by 397 cu. in. Find the edges of each cube. 10. A rectangular sheet of tin, 12 in. by 16 in., is made into an open box by cutting out a square from each corner and turning up the sides. Find the size of the square cut out if the volume of the box is 180 cu. in. 16" 11. A rectangular piece of tin a inches longer than it is wide is made into an open box, containing c cubic inches, by cutting from each corner a square of side b inches. Find the dimensions of the original piece of tin. 12. One edge of a rectangular box is increased 6 in., another 3 in., and the third 4 in., making a cube whose volume is 862 cu. in. greater than that of the original box. Find the dimensions of the box. Ans. 4.14 in. by 7.14 in. by 6.14 in. 13. If in a right prism the altitude is equal to a side of the base, find the volume if the base is an equilateral triangle whose sides are a. 14. Find the capacity in gallons of a water tank for a locomotive tender. The dimensions are as given in the figure for the length cross section, and the width is 9 ft. 6 in. 5 k 15. A certain coal and coke company finds it necessary to construct a wharf wall 150 ft. long, and having a cross section with dimensions as shown in the figure. Find the number of cubic yards in the wall. By "batter 1:12" is meant that the slant is 1 ft. in a I vertical rise of 12 ft. 16. In a right prism whose volume is 54, the lateral area is four times the area of the base which is an equilateral triangle. Find the edge of the base. Ans. 6. 13" F -22 Batter 1:12 91811 1'6" 15' 17. The total areas of two similar cylinders of revolution are 75 sq. in. and 192 sq. in. respectively. If the volume of the first cylinder is 250 cu. in., what is the volume of the second? 18. In a right circular cylinder, given V=πr2h_and_T=2πr2+2πrh. Find T in terms of V and h. Ans. T = 2V +2√√πh. 19. In a right circular cylinder, find V in terms of the circumference c of the base and the total area T. Ans. V = 20. Find the steam capacity of a horizontal cylindrical boiler 4 ft. in diameter and 16 ft. long, if the height of the segment occupied by the steam is 18 in. (Use (2) of §509). 21. A cylindrical tank of diameter 30 in. and 34 in. long rests on its side. Find the number of gallons of gasoline in the tank if the depth is 5 in. Ans. 13 gal. nearly. 22. The volume of an irregular shaped body is often found by immersing it in water and determining the amount of water displaced. A cylindrical vessel that has a diameter of 4 in. is partly filled with water. A stone immersed in the water raises its level 31⁄2 in. Find the volume of the stone. 23. A silo is used to keep fodder in a green and succulent state for feeding farm animals. It is usually built in a cylindrical form. Find the capacity in tons of a silo in the form of a right circular cylinder 20 ft. in diameter and 32 ft. high, if a cubic foot of silage weighs 40.7 lb. 24. An 18 acre field yields 11.5 tons of silage per acre. What must be the height of a cylindrical silo 20 ft. in diameter to hold all the silage if 1 cu. ft. of silage weighs 38.4 lb.? 25. A silo is in the form of a right circular cylinder, and is 20 ft. in diameter inside and 32 ft. high. How many cubic yards of concrete did it take to build it, if the floor and wall are each 6 in. thick, and the foundation wall is 8 in. thick and 5 ft. deep? *5% 26. The connecting rod with dimensions as given in the figure is made from stock with the dimensions shown. Find K 18 length of stock that it is necessary Connecting Rod to allow for the cylindrical part of the rod. " Stock k-is 27. Find the volume of the beveled washer with dimensions as shown in the figure. 28. A pine log 2 ft. in diameter and 16 ft. long is floating in water. Find the weight of the log if two-thirds of the volume of the log is under water. (Water weighs 62.5 lb. per cubic foot.) ··3% -8% 29. Will a floating pine log 11⁄2 ft. in diameter and 10 ft. long support a man weighing 180 pounds if the specific gravity of the log is 0.72? 30. The following is the record of a test in which a high speed drill removed 70.56 cu. in. of cast-iron per minute. The penetration per minute was 571⁄2 in., the feed in. per turn, and the number of turns per minute was 575. Do these numbers agree? |
