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2 O feet, the solidity. Er. 2. What is the solidity of a joist 3 in. by gin. and Jof. long?
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Prob. 2. To measure timber-trees, or unsquared timber, equally unck.
Method 1. Multiply the square of of the tree's compass for the side of a square equal in area to the end of the tree, by the length of the tree, and the product will give the solidity.
This method, though easy in practice, is very erroneous in principle, as the content by this rule is too small by above one-fourth of itself.
The true rules for measuring round timber, have been already given for measuring a cylinder : but if this rule should be thought troublesome, the following is a method which will come very near the truth, and nearly as expeditious in practice as the above method, and therefore may be esteemed true.
Method 2. Multiply the square of one-fifth of the tree's compass by the length of the tree, and double the product will be the content.
MISCELLANEOUS QUESTIONS. 1. What difference is there between a floor 48 feet long, and 30 teet broad, and two others each of balf the dimensions ?
Ans. 720 feet. 2. From a mabogany plank 26 inches broad, a yard and a half is to be sawed off; at what distance from the end must the line be struck ?
Ans 6 23 feet. 3. The sides of three squares being 4, 5, and 6 feet, respectively, it is required to find the side of a square that shall be equal in area io all the three.
Ans. 8°7749 feet. 4. What quantity of canvas will be necessary for forming a conical tent, whose height is 8 feet, and the diameter at bottom 13 feet?
Ans. 210. square yards.
3. How many square feet of board are required to make a rectangular box, whose length is to be 34 feet, breadth 2 feet, and depth 20 inches :
Ans. 32 feet. 6. A joist is 8 inches deep, and 3} broad ; what will be the dimensions of a scantling just as big again as the joist, that is 4 inches broad ?
Ans. 12:52 inches deep. 7. A roof which is 24 feet 8 inches by 14 feet 6 inches, is to be covered with lead at 8 lbs to the foot ; what will it come to at 18s.
Ans. 221. 195. 10:d. 8. If the side of an equilateral triangle be 10 chains, what will be the side of another equilateral triangle, whose area is one-fourth of the former ?
Ans. 5 chains. 9. What is the side of that equilateral triangle whose area cost as much paving at 8d. per foot, as the pallisading the three sides did at a guinea a yard?
Ans. 72-746 feet. 10. What would a circular reservoir, whose diameter at top is 40 yards, at bottom 389 yards, and its side, or slant depth 11 feet, cost lining with brick-work, at 3s. 10d. the square yard ?
Ans. 3111. 185. 2d. 11. The four sides of a field, whose diagonals are equal to each other, are 25, 35, 31, and 19 poles, respectively; what is the area?
Ans. 4 acres, 1 rood, 38 poles. 12. What is the length of a cord that will cut off one-third of the area from a circle whose diameter is 289 ?
Ans. 278 6716. 13. A cable which is 3 feet long, and 9 inches in compass, weighs 22 lbs. what will a fathom of that cable weigh whose diameter is 9 inches ?
Ans. 434.26 lbs. 14. A circular fish-pond is to be dug in a garden, that shall take up just half an acre ; what must the length of the chord be that strikes the circle ?
Ans. 27 75 yards. 15. A carpenter is to put an oaken curb to a round well, at 8d. per square foot; the breadth of the curb is to be 7 inches, and the diameter within 31 feet; what will be the expense : Ans. 5s. 21d.
16. Suppose the expense of paving a semicircular plot, at 25. 4d. per foot, amounted to 101. what is the diameter of it ? Aus. 14.7737.
17. Seven men bought a grinding-stone of 60 inches in diameter, each paying one-seventh part of the expense; what part of the diameter must each grind down for his share ? Ans. first, 4:4503 ; second, 4 8400 ; third, 5.3535; fourth, 6 0765 ; fitth, 7:2079 ; sixth, 9 3935; and the seventh, 22-6778.
18. A garden 100 feet long, and 80 feet broad, is to have a gravel walk of an equal width half round it; what must the width of the walk be so as to take up just half the ground? Ans. 25.968 feet.
19. How many galions, wine measure, will a cistern hold, supposing its length and breadth at top to be 5 and 4 feet respectively, and ai bottom 4 and 3 feet; the perpendicular depth being 34 feet?
Ans. 414, gallons. 20. A malster has a kiln that is 16 feet 6 inches square, which he
wants to pull down, and to build a new one that will dry three times as much at a time as the old one ; what must be the length of its side ?
Ans. 28 feet 7 inches. 21. If a round cistern be 26'3 inches in diameter, and 52.5 inches deep, how many inches diameter must a cistern be to hold twice the quantity, the depth being the same?
Ans. 37.19 inches. 22. A may-pole, whose top was broken off by the wind, struck the ground at 15 feet distance from the bottom of the pole; what was the height of the whole may-pole, supposing the length of the broken piece to be 39 feet?
Ans. 75 feet. 23. What will the diameter of a globe be, when its solidity and superficial content are equal to each other, or rather when they are both expressed by the same number?
Ans. 6. 24. How many three-inch cubes can be cut out of a 12-inch cube?
Ans. 64. 23. A farmer borrowed part of a hay-rick of his neighbour, which measured 6 feet every way, and paid him back again by two equal cubical pieces, each of whose sides were three feet. Query, whether the lender was fully paid ?
Ans. He was paid & part only. 26. What will the painting a conical church spire come to at 8d. per yard; supposing the circumference of the base to be 64 feet, and the altitude 118 feet ?
Ans. 141. Os. 8fd. 27. What will a marble frustum of a cone come to at 12s. per solid foct; the diameter of the greater end being 4 feet, that of the less end 14 feet, and the length of the slant side 8 feet ?
Ans. 301. ls. 10d. 28. The diameter of a legal Winchester bushel is 184 inches, and its depth 8 inches; what must the diameter of that bushel be whose depth is 7į inches?
Ans. 19:10671. 29. Three men bought a tapering piece of timber, which was the frustum of a square pyramid ; one side of the greater end was 3 feet, one side of the less I foot, and the length 18 feet ; what must be the length of each man's piece, supposing they paid equally, and are to have equal shares?
Ans. first, 3.269 ; second, 4.559 ; and the third, 10*172 ;
reckoning from the greater end to the less. 30. Supposing the ball at the top of St. Paul's Church to be 6 feet in diameter; what would the gilding come to at 3 d. per square inch?
Ans. 2371. 10s. Id. 31. A person wants a cylindric vessel of 3 feet deep, that shall hold twice as much as another of 28 inches deep, and 46 inches in diameter ; what must be the diameter of the required vessel ?
Ans. 57.37 inches. 1. 32. Two porters agreed to drink off a quart of strong beer between them, at two pulls, or a draught each ; now, the first having given it a black eye, as it is called, or drank till the surface of the liquor touched the opposite edge of the bottom, gave the remaining part of it to the other; what was the difference of their shares, supposing the pot to be the frustum of a cone, the depth of which is 5-7 inches, the diameter at the top 3.7 inches, and that of the bottom 4-38 inches ?
Ans. 7°07 cubic inches. 33. Three persons having bought a sugar-loaf, want to divide it equally among them by sections parallel to the base; it is required to find the altitude of each person's share, supposing the loaf to be a cone, whose height is 20 inches ? Ans. 13.867 the upper part,
3.604 the middle part, and 2-528 the lower part. 34. How high above the surface of tht earth must a person be raised to see a third part of its surface?
Ans. To the height of the earth's diameter. 35. A cubical foot of brass is to be drawn into a wire of 1-40ub of an inch in diameter ; what will be the length of the wire, allowing po loss in the metal ?
Ans. 97784-797 yards, or near 56 miles. 36. A bowling-green, 300 feet long, and 200 feet broad, is to be ra:sed one foot higher, by means of the earth to be dug out of a diich that goes round it; to what depth must the ditch be dug, supposing its breadth to be every where 8 feet?
Ans. 7 feet. 37. Of what diameter must the bore of a piece of ordnance be, which is cast for a ball of 24 lbs. weight, so that the diameter of the bore may be 1-10th of an inch more than that of the ball ?
Ans. 5.757 inches. 38. If a sphere of copper, of one foot in diameter, was to be beat out into a circular plate of 1-20th of an inch thick, what would be its diameter ?
Ans. 12 04 feet. 39. The perambulator, or surveying wheel, is so contrived as to turn just twice round, in the length of a pole, or 16. feet; what is
a its diameter ?
Ans. 2.626 feet. 40. The ellipse in Grosvenor-square measures 840 links across the longest way, and 612 the shortest, within the rails : now the walls being 14 inches thick, it is required to find what ground they inclose, and what they stand upon ? Ans. They inclose 4 acres, o rood,
6 poles, and stand on 17604 square feet. 41. If a heavy sphere, whose diameter is 4 inches, be put into a conical glass, full of water, whose diameter is 5, and altitude 6 inches; it is required to find how much water will run over?
Ans. of a pint nearly. 42. Supposing it to have been found, by measurement, that a magof-war, with its ordnance, rigging, and appointments, displaces 50,000 cubic feet of water ; what is the weight of the vessel ?
Ans. 1395 36
tons. 13. Supposing it were required to make a vessel of a foot deep, in the form of a frustum of a cone, that shall hold 13 ale gallons, and have its top and bottom diameters in proportion to each other, as 5 is to 3 ; what must be its dimensions ? Ans. The bottom diameter is 14:64017, and the top diameter 24:40028.