14. A man being asked how many sheep he had, said, if he had as many more, half as many more, and 7 should have 20; how many had he? 15. In an orchardthe trees bear apples, plums, and 50 of them cherries; how many trees in all? sheep, he Ans. 5. pears, are there Ans. 600. 16. A can do a piece of work alone in 10 days, B can do it in 13; in what time will both together do it? Ans. 515 days. 17. What is the difference between the interest of £350 at 4 per cent. for 8 years, and the discount of the same sum at the same rate, and for the same time? Ans. £27 38. 18. Sound moves at the rate of 1142 feet in a second; if the time between the lightning and thunder be 20 seconds, what is the distance of the explosion? Ans. 4.32+miles. 19. If the earth's diameter be 7911 miles, and that of the moon be 2180, how many moons will be required to make one earth? Ans. 47.788+. 20. If a cubic foot of iron were drawn into a bar of an inch square, what would be its length, supposing no waste of metal? 12×12×12_27648in.-2304ft. Ans. .25X.25 21. A lent B a solid stack of hay, measuring 20 feet every way; some time after B returned a quantity measuring every way 10 feet; what proportion of the hay is yet due? Ans. . 22. A general disposing his army into a square, finds he has 284 soldiers over and above, but increasing each side by one soldier, he wants 25 to fill up the square; how many soldiers had he? Ans. 24000. 340. 23. Supposing a pole 75 feet high to stand on a horizontal plane, at what height must it be cut off, so as that the top of it may fall on a point 55 feet from the bottom, and the end where it was cut off, rest on the stump or upright part? 75X75-55X55 71 ft. Ans. 75X2 RULE. From the square of the length of the pole (i. e. the sum of the hypothenuse and perpendicular) take the square of the base; then divide the remainder by twice the length of the pole, and the quotient will be the height at which it must be cut off. 24. Suppose a ship sail from lat. 43° N. between N. and E. till her departure from the meridian be 45 leagues, and the sum of her distance and difference of latitude be 135 leagues; what is the distance sailed, and the latitude come to? 135X135-45X45 lea. m. 135-60-751. dis. s'd. =60-180-3° of lat. 43°+3°46° come to. 135×2 Ans. 341. 25. Four men bought a grindstone 60 inches in diameter; how much of its diameter must each grind off to have an equal share of the stone, if one grind his share first, and then another, till the stone is ground away, making no allowance for the eye? RULE.-Divide the square of the diameter by the number of men, subtract the quotient from the square, and extract the square root of the remainder, which is the length of the diameter, after the first share is taken off; and by repeating the latter part of the process, all the several shares may be found. 60×60÷4=900, the subtrahend. 3600—900—51.96+ and 60—51.96—8.04, 1st share. /2700--900—42.42+ and 51.96—42.42—9.54, 2d share. 1800-900-30. and 42.42-30-12.42, 3d share. and 30, 4th's share. 26. Suppose one of those meteors called fireballs to move parallel to the earth's surface, and 50 miles above it, at the rate of 20 miles per second; in what time will it move round the earth? The earth's diameter being 7964 miles, the diameter of the orbit will be 7964+50×2=8064, and 8064×3.1416—25333.8624, its circumference. Then 25333.8624-20-1266.69312s.-21 6" 41" 35" 13" 55""""" Ans. 27. When first the marriage knot was tied betwixt my wife and me, What were our ages at marriage? Ans. 57 and 33. 28. A body weighing 30 lb. is impelled by such a force as to send it 20 rods in a second; with what velocity would a body weighing 12 lb. move, if it were impelled by the same force? Ans. 50 rods. 29. In a thunder storm I observed by my clock that it was 6 seconds between the lightning and thunder; at what distance was the explosion? Ans. 6852 ft. 1121 miles. 44 30. There is a square pyramid, each side of whose base is 30 inches, and whose perpendicular height is 120 inches, to be divided into three equal parts by sections parallel to its base; what will be the perpendicular height of each part? 30X30X40 36000, the solidity in inches. Now of this is 24000, and is 12000. Therefore, as 35000: 120×120×120 1152000 :: {12000: 576000} Then, √31152000=104.8. Also, 576000-83.2. Then 120-104.8-15.2, length of the thickest part, and 104.8-83.2-21.6, length of the middle part: consequently, 83.2 is the length of the top part. 31. I have a square stick of timber 18 inches by 14, but one with a third part of the timber in it, provided it be 8 inches deep, will serve; how wide will it be? Ans. 10 inches. 32. There are 4 spheres, each 4 inches in diameter, lying so as to touch each other, in the form of a square, and on the middle of this square is put a fifth ball of the same diameter; what is the distance between the two horizontal planes passing through the centres of the balls? No42+42÷÷22.828+ inches, Ans. 33. There are 2 balls, each 4 inches in diameter, which touch each other, and another ball of the same diameter is, so placed between them that their centres are in the same vertical plane; what is the distance between the horizontal planes which pass through their centres? /42-22-3.46+in. Ans. 34. A military officer drew up his soldiers in rank and file, having the number in rank and file equal; on being reinforced with three times his first number of men, he placed them all in the same form, and then the number in rank and file was just double what it was at first; he was again reinforced with three times his number of men, and after placing the whole in the same form as at first, his number in rank and file was 40 men each; how many men had he at first? Ans. 100. 35. If a weight of 1440 lb. be placed 1 foot from the prop, at what distance from the prop must a power of 160 lb. be applied to balance it? Ans. 9 feet. 36. Three men wishing to carry a stick of timber, which is of uniform size and density, and 30 feet long; if one man takes hold at one end of the stick, how far from the other end must the other two take hold together, that each may bear an equal portion? Ans. 7 feet. The centre of gravity being in the middle of the stick, we may regard its weight as all accumulated in that point, and the stick itself as a lever supporting it; and then the parts borne will be inversely as the distances from the middle, and the reverse, i. e. the man at the end being 1,5 feet from the middle, the 2 must be of 15, or 7.5 feet from the middle, and 15-7.5-7.5=the distance from the end. Where ought the 2 men to take hold in order to carry of the stick? The one being 15 feet from the middle, the two, in order to carry 3 times as much, must be 1-3d of 15-5 feet from the middle, and 15—510 ft., the distance from the end. 37. Suppose a pole 100 feet high, to be 24 inches in diameter at the ground, and 4 in. do. at the top, and a vine 1 inch in diameter at the ground to run up this pole, winding round every 3 feet, and gradually diminishing so as to come to a point at the top of the pole, what is the length of the vine? Ans. 162 feet, 11.94 inches. SECTION V. PRACTICAL RULES AND TABLES. 342. MEASURES OF CAPACITY. The English Winchester bushel, containing 2150.4 cubic inches, or 77.6274 lb. avoirdupois, of pure water, at its maximum density, is established, at the custom-houses in the United States, as the standard of dry measure; and the wine gallon of 231 cubic inches, or 8.339 lb. of water, as above, is established as the standard of liquid measure. The above are also the measures established by law in Vermont and some other states. But in New York, according to their revised laws, the legal bushel contains 2211.84 cubic inches, and the liquid gallon 221.18 cubic inches. In measuring coal, lime, ashes, and some other articles, it is customary to use a larger measure. In Vermont the bushel, for these articles is established by law at 38 quarts, of which the common bushel holds 32, but in most places the bushel for coal, &c. contains 40 quarts. 1 cubic foot-0.80356 bush. Winchester measure. 1 cubic foot 0.67669 bush. Vermont coal, &c. measure. 1 cubic foot 0.64285 bush. com. coal, &c. measure. 343. To find how many bushels any bin, box, or coal-house will contain. RULE. Find the content in feet, and multiply it by the decimal of a bushel standing against 1 cubic foot in the above table. EXAMPLES. 1. The dimensions of a coal-box were length 12.5 ft., height 3.4 ft., width at the top 3.94 ft., width at the bottom 2.7 ft.; how many bushels of each of the above. measures will it hold? 3.94+2.7-6.64, and 6.64-2-3.32, and 3.32×3.4×12.5=141.1 cubic feet. Then 141.1X0.8 112.88 bush. Win. 141.1X0.677 95.52 bush. Vt. coal meas. 90.30 bush. com. coal meas. 2. If a coal-house be 50 feet long, 40 feet wide, and 20 feet high, how many bushels will it hold? 50X40X20 40000 cu. ft., and 40000×0.67669-27067 b.Vt. m. 50X40X20 40000 cu. ft., and 40000X0.64285-25714 b. c. m. 344. Having two dimensions in feet of a bin, box, or coal-house, to find what the other must be in order to hold a given quantity. RULE.-Multiply the given dimensions together for a divisor, and multiply the given quantity by the cubic feet in a bushel, as expressed in the above table; the quotient will be the other dimension. 1. A coal-box is 25 feet wide and 4 feet long; how high must it be to hold 10 bushels? 2.5×4=10 divisor, 10×1.4777=14.777 & 14.777÷10=1.4777 ft.=1ft. 5gin. 2.5×4=10 divisor, 10×1.5555—15.555 & 15.555÷10—1.5555 ft.=1 ft.6 in. 2. If I build a coal-house 40 feet wide and 18 feet high, how long must it be to hold 30000 bushels common coal measure? Ans. 64.81 feet. 3. I have a garner of wheat which is 20 feet long, 8 feet wide, and 6 feet high; how many bushels are there? Ans. 20X8X6X0.8-768 bushels. 4. How high must the above garner be to hold 1000 bushels of wheat? Ans. 20×8-160 for a divisor, and 1000×1.2444—1244.4 for a dividend. Then 1244.4-160-7.77 feet, for the height of the garner. |12|0.7854||19|1.9689||26|3.6863||33|5.9395||40| 8.7179||47|12.0482| 13 0.9218 20 2.1817||27|3.9753||34|6.3050 41 9.1684||48 12.5664 14 1.0691 21 2.4048 28 4.2760 35 6.6813|42 9.621149 13.0954 |151.2272||22|2.6393||29|4.5869||36|7.0686 43 10.0847 5013.6354 16 1.3963 23 2.8847 30 4.9087 37 7.4667 44 10.5592|51|14.1861 171.5762 24 3.1416 31 5.2414 38 7.8758 45 11.0447 52 14.7479 181.7671 25 3.408232 5.5851||39|8.2957|46|11.541053 15.3201 The column marked diameter is the diameter in inches, and the column marked area is the area of a section of the cylinder in feet and decimal parts. To illustrate the use of this table, I will give a few examples, viz. 1. How many cubic feet in a round stick of timber, 20 feet long, and 18 inches diameter ? Look in the table under the head of diameter, and against 18 in the column of areas is 1.7671, which multiplied into the length in feet, will give the number of cubic feet such stick contains that is, 1.7671×20=35.342 cubic feet. 2. How many cubic feet in a round log 24 inches diameter and 16 feet long? Ans. 3.1416x16 50.2656 cubic feet. 3. Suppose the mean diameter of a cask to be 3 feet, and its length 5 feet, how many cubic feet will it contain, and how bushels of wheat will it hold? many Ans. 7.0686X5=35.343 cubic ft., which X0.8-28.2744 bush. |