6. What is the surface of an oblique regular hexagonal prism, a side of the base being 10 inches, its length 20 feet, and its perimeter perpendicular to its lateral edges 4 feet? A cylinder may be regarded as a prism of an infinite number of sides, the ends being circles. To find the solidity of a cylinder: Multiply the area of the end, or base, by the length, or altitude. Or, multiply the length by double the square of onefifth of the circumference, and the product will be the solidity nearly. EXAMPLES. 1. What is the solidity of a cylinder 20 feet long and 24 inches in diameter ? Ans. 62.832 cubic feet. For the method of finding the gauge points for the cylinder, see examples under the circumscribing and inscribed square, 22. To solve the above example by the sliding rule:Place the length of the cylinder in feet over 13.54 on D, and over the diameter in inches found on D, will be found the solidity in cubic feet on the line C. 2. If a square prism be hewn or cut from a cylinder 20 feet long and 24 inches in diameter, what will be the side of the square? and how many solid feet will the prism contain? Answers, 16.97 inches; and 40 feet. 3. What is the content of a cylinder 55 feet in length and 20 inches in diameter ? Ans. 120 feet, nearly. 4. Find the solidity of a cylinder, the height of which is 25 inches, and its diameter 15 inches. 5. The circumference of the base of 20 feet, and its perpendicular height is volume ? Ans. 2.5566 feet. an oblique cylinder is 19.318; what is its Ans. 614.93 feet. 6. How many bushels will a cylinder contain, which is 7 feet long and 30 inches in diameter? Ans. 27 bush. nearly. The gauge point for bushels, if the length or altitude of the cylinder be taken in inches, is 52.32 on D; or, if the length be taken in feet, the gauge point is 15.11. Therefore, place 7 over 15.11 on D, and over the diameter, 30 inches, found on D, will be found the number of bushels on C. 7. A bushel measure is 18.5 inches in diameter and 8 inches deep; how many solid inches does it contain? Ans. 2150.42 inches. 8. How many wine, and how many ale gallons will a round vessel hold, whose diameter is 30 inches, and depth 23 inches? Answers, 54 wine, and 444 ale gallons. The gauge points for this example will be found in 22, examples 10 and 11. 9. How many barrels of wine and how many of ale will a cistern hold, whose diameter is 70 inches and altitude 60 inches? Answers, 31.75 wine, and 22.8 ale, nearly. For the gauge points for wine and ale barrels, see 22, examples 13 and 14. 10. What must be the length of a cylinder 30 inches in diameter in order that it may hold a barrel of wine? and what its length in order that it may contain a barrel of ale ? Answers, 10.29 and 14.50 inches, nearly. 11. A cylinder contains 44 cubic feet, and its diameter is 30 inches; required its length. Ans. 9 feet, nearly. 12. What must be the length of a cylinder 20 inches in diameter, in order that it may contain 40 ale gallons? Ans. 35.87 inches. To solve the last three examples by the sliding rule :—Place the contents over the given diameter in each example, and over the respective gauge points will be found the answers required. Thus, place 311, the number of wine gallons in a barrel, over 30 on D, and over 17.15, the gauge point for wine gallons, will be found 10.29 inches, the answer required; and place 36, the number of ale gallons in a barrel, over 30 on D, and over 18.95 will be found 14.50 inches; and place 44 over 30, and over 13.54, the gauge point for round timber, will be found 9 feet. 13. How many solid feet, how many bushels, and how many wine hogsheads, of 63 gallons each, will a cylinder hold, whose diameter is 18 inches, and length 15 feet? Answers, 26.6 feet; 214 bushels; 3.15 hogsheads. The gauge point for a wine hogshead, the length being taken in feet and decimal parts of a foot, is 39.29 inches. T87. MILL LOGS. TO GAUGE MILL LOGS, THE DIAMETER OF A STANDARD LOG BE ING GIVEN : Place 1 on C (calling the 1 one log) over the diameter of the standard log on D; then over the diameter of any log found on D, will be found its ratio to the standard log. In some states, a log 19 inches in diameter is called a standard log; and in some states, a log 22 inches in diameter is called a standard log; and in some, 24 inches is the established diameter. EXAMPLES. 1. Calling 19 inches the diameter of a standard log, it is required to find the ratio of the following logs to the standard log, (that is, to find what they will count in buying and selling;) one 14 inches in diameter, one 12, one 91⁄2, one 17, one 25, one 38, and one 60 inches in diameter. Answers in order,-.542; 4; ; .8; 1.72; 4; and 10 logs. This method gives the exact ratio of the several logs to the standard log. Thus, a log 38 inches in diameter counts 4, its solid contents being exactly 4 times greater than the contents of a log 19 inches in diameter; and a log 60 inches in diameter counts 10, or 9.97, its solid contents being nearly 10 times that of a log 19 inches in diameter. The value of logs designed for boards, does not, however, depend wholly on the number of cubic feet which they contain, for the reason that wide boards are more valuable than narrow ones in proportion to the number of feet which they contain, and likewise, because there is generally less loss of timber in sawing large logs into boards than small ones, in proportion to their cubic contents. 2. Assuming the diameter of a standard log to be 22 inches, find the ratio of the following logs to the standard: one 20 inches, one 30 inches, and one 37 inches in diameter. Answers, .82; 1.85; and 2.9. 3. If a standard log, 19 inches in diameter, be valued at 90 cents, what will be the value of the following logs: one 15 inches in diameter, one 20 inches, one 25 inches, and one 31 inches in diameter ? Answers in order,-$0.56; 0.997; 1.56; and 2.40. Place the value of the standard log, found on C, over its diameter on D, and over the diameter of any log found on D, will be found its value on C. A log 13 feet in length and 19 inches in diameter at the smaller end, will make 200 feet of inch boards; therefore, if we place 200 on C, over 19 on D, over the diameter of any other log, (taken in inches, and of the same length,) will be found nearly the quantity of inch boards which may be cut from it. 4. How many feet of inch boards will each of the following logs make, the length of each being 13 feet, and their diameters, 15 inches, 20 inches, 22 inches, and 28 inches ? Answers, 123; 221; 267; and 450 feet. To find the number of feet of inch boards which any log will make : If the log be 2 feet in diameter, or less than two feet, allow 2 inches on four sides for the thickness of the slabs, and onefifth for saw-calf, and 1 board for wane; but if the log is more than 2 feet in diameter, allow 3 inches for the thickness of each of the four slabs, and one-fifth for saw-calf, and 2 boards for wane. If, however, the logs are very straight and smooth, may be thinner. the slabs EXAMPLES. 1. How many feet of inch boards will a log make whose length is 16 feet, and diameter, at the smaller end, 20 inches? Ans. 255 feet. four sides it will be 16 of 16 for saw-calf, and After the log has been slabbed on inches thick; then, from 16 subtract the remainder is 13 nearly, which is the number of boards the log will make; and allowing 1 board for wane, there will be 12 boards left, each 16 inches wide, and each containing 214 square feet consequently, the log will make 255 feet of boards. 2. How many feet of inch boards in a log 16 inches in diameter, and 14 feet long? Ans. 112 feet. 3. How many feet of boards will a log make, its length being 13 feet, and diameter 31 inches? Ans. 487 feet. See the rule for gauging boards, ¶ 15, example 4. Market boards are usually a little less than one inch in thickness; and consequently, the number of feet of market boards in a log will be greater than the number of feet of inch boards. To find the number of feet of market boards, in any log, allow one-eighth for saw-calf, and apply the above rule for inch boards with this difference. 4. How many feet of market boards will a log make, its length being 20 feet, and its diameter 46 inches? Ans. 2,200 feet. The following table exhibits the number of feet of market boards in any log, whose diameter at the smaller end ranges from 15 to 36 inches, and its length from 10 to 15 feet. |