27. The versed sine of a circular arc is 2 feet, and the diameter of the circle 18 feet; required the length of the chord which cuts off the arc. Ans. 11.3136 feet. 28. How many perches of stone will be required to build a wall 2 feet thick and 4 feet high, around a circular garden which contains half an acre? Ans. 288.3 perches. 29. The area of a rectangle is 10 acres, 3 roods, and 8 rods, and its diagonal is 60 rods; required the sides. (See 65, problem 13.) Ans. 48 and 36 rods. 30. What is the area of a circular segment, its chord being 30, and versed sine 5 rods? Ans. 102.187 rods. 31. Required the area of a circular zone, the chord of one of the segments being 8 rods, the versed sine of the other segment 4.8 rods, and the diameter of the circle 12 rods. Ans. 62.5 rods. 32. Suppose a tree 100 feet in height to be broken off by the wind, and that the top of the tree strikes the ground 40 feet from its base, whilst the other end of the part broken off rests on the top of the stump; required the length of the part broken off. (See 65, problem 1.) Ans. 43 feet. 33. There are two parallel walls standing on a plane, and the foot of a ladder, whose length is 263 feet, may be so planted that it will just reach the top of each wall; one of the walls is 22 and the other 14 feet in height: I require their distance from each other. Ans. 37 feet and 9.24 inches. 34. Required the side of the greatest square stick, the side of the greatest equilateral triangular prism, and also the side of the greatest octagonal prism, which can be cut from a round log, whose diameter at the smaller end is 18 inches. Ans. in order,-12.73; 15.588; and 6.5 inches. 35. Required the side of an octagon cut from a square 2 feet on a side. Ans. 10 inches, nearly. 36. Required the diameter of a circle, in which a decagon 4 inches on a side may be inscribed. Ans. 12.945 inches. 3 37. Required the area of a crescent, whose chord is 24, and the versed sines of the arcs 2 and 10. Ans. 148.2374. 38. What is the area of an ellipse, whose major and minor axes are 60 and 40 rods? Ans. 11.781 acres. 39. What is the circumference of an ellipse, whose axes are 10 and 100? Ans. 204, nearly. 40. The area of a circle is 4 acres less than the area of its least circumscribing square; required its diameter. Ans. 13.6525 chains. 41. A log of wood is 15 inches broad and 11 thick; what length of it will make 10 cubic feet? Ans. 8 ft. 8 inches. 42. A round cistern is 26.3 inches in diameter; what must be the diameter of another to contain twice as much, its depth being the same? Ans. 37.19 inches. 43. What will be the expense of painting a conical church spire, at 8d. per yard, the circumference of the base being 64 feet, and its slant height 118 feet? Ans. £13 19s. 84d. 44. What will be the expense of gilding a sphere, 6 feet in diameter, at 6 cents the square inch? Ans. $1,017.878. 45. How many three-inch cubes can be cut out of a cubic foot? Ans. 64. 46. The numbers expressing the solidity and the surface of a sphere are the same; what is its diameter? Ans. 6. 47. How far will a point in the circumference of a wagonwheel move whilst the wagon is drawn one mile over a smooth level road, the diameter of the wheel being 4 feet? Ans. 6,722.688 feet. 48. If two wheels, whose diameters are 4 and 5 feet, be placed on the ends of an axletree 20 feet long, and set rolling on a plane, what will be the diameter of the circle described by the larger wheel? Ans. 200 feet. 49. How far may the lamp of a lighthouse, 150 feet high, be seen at sea, when the eye of the observer is elevated 100 feet above the surface of the water? Ans. 30 miles, nearly. 50. What is the side of an equilateral triangle, whose area cost as much for paving, at 8d. per foot, as the fencing of the three sides at a guinea a yard, the value of a guinea being 21 shillings? Ans. 72.746 feet. 51. There are two round logs of the same length, and their diameters are 2 and 5 feet; how much more timber is contained in the larger than in the smaller log? Ans. 6 times as much. 52. Required the side of a cubic box, whose capacity is 10 bushels. Ans. 27.808 inches. 53. What is the length of a diagonal between the opposite corners of a cube whose side is 20 inches ? Ans. 34.641 inches. 54. How many bushels will a bin hold, its length being 5 feet 6 inches, its breadth 4 feet 9 inches, and its depth 3 feet 9 inches? Ans. 78.724 bushels. 55. The side of an equilateral triangular prism is 18 inches, and its solidity 39 feet; required its length. Ans. 40.029 feet. 56. A round cistern is 5 feet 6 inches in diameter; what must be its depth to hold 20 wine hogsheads? Ans. 7 feet 2 inches, nearly. 57. How many bricks will it take to build a wall 10 feet high, and 250 feet long, if the bricks be 10 inches long, and 4 courses make one foot in height, the breadth of the wall being equal to the breadth of 3 bricks? Ans. 36,000. 58. The same number expresses the solidity and convex surface of a cylinder; what is its diameter ? Ans. 4. 59. A gentleman has a garden 100 feet long and 80 feet broad; required the breadth of a walk extending round two sides, which shall cover half the area. Ans. 25.968 feet. 60. What must be the diameter of a bushel measure, depth being 7 inches? Ans. 19.1067 inches. its 61. The ditch around a fortification is 1,000 feet long, 9 feet deep, and 20 feet broad at the bottom and 22 at the top; how many ale gallons of water will fill the ditch? Ans. 1,158,128. 62. The solidity of a cone is 20 feet, and the diameter of its base 20 inches; required its length. Ans. 27 feet. 63. The altitude of a cone is 10 feet, and its solidity is equal to 21 bushels; required the diameter of its base. Ans. 37.9137 inches. 64. How many wine, and how many ale hogsheads will a cistern hold, whose diameters at the top and bottom are 42 and 5 feet, and its depth 8 feet? Ans. 17.734 wine, and 16.9 ale hhds.. 65. What is the capacity of a box whose dimensions are, at the top 7 and 6 feet, at the bottom 5 and 3 feet, and its depth 4 feet, its shape being that of a prismoid, or frustum of an indirect wedge? Ans. 110 feet. 66. Find the solidity of a prismoid, the length and breadth of its greater end being 24 and 16 inches, those of the less 16 and 12 inches, and its length 10 feet. Ans. 19.624 feet. 67. Required the contents of a sphere in feet and bushels, its diameter being 2 feet 6 inches. Ans. 8.179 feet; 6.58 bushels. 68. The expense of gilding a ball at $1.80 per square foot is 40 dollars; required its diameter. Ans. 2.6596 feet. 69. If a cone 40 inches high is cut into three equal parts by planes parallel to the base; what must be their lengths? Ans. 27.734; 7.21; and 5.056 inches. 70. The frustum of a square pyramid is 18 feet long, and the sides of its ends are 1 and 3 feet, and it is divided into three equal portions; what must be the length of each ? Ans. 3.269; 4.559; and 10.172 feet. 71. The solidity of the frustum of an equilateral triangular pyramid is 4 feet, and the sides of its ends 14 and 8 inches; required its length. Ans. 10.727 feet. 72. Required the diameter of a spherical vessel whose capacity is 1 bushel. Ans. 16.04 inches. 73. What is the side of the greatest cube that can be cut from a sphere 18 inches in diameter ? Ans. 10.392 inches. 74. What must be the diameter of a cylindric vessel 3 feet deep, that will contain twice as much as another 28 inches deep and 46 inches in diameter? Ans. 57.37 inches. 75. A cubic foot of brass is to be drawn into a cylindric wire of an inch in diameter; what will be the length of the wire? Ans. 97,784.797 yards. 76. A bowling-green 300 feet long and 200 broad, is to be raised one foot higher by means of earth dug from a ditch to be made round it; what must be the depth of the ditch, its breadth being 8 feet? Ans. 78 feet. 77. A circle 60 inches in diameter is to be divided into three equal portions by means of two concentric circles; what must be their diameters? Ans. 34.641; and 48.988 inches. 78. The diameter of a circle is 20 rods; what is the area of the inscribed square? Ans. 200 rods. 79. The diameter of a sphere is 12 inches; it is required to find the diameter of another three times as large. Ans. 17.307 inches. 80. Required the weight of a bushel of water. Ans. 77.778 lbs. 81. Required the weight of an oak stick 12 feet long and 31 inches square. Ans. 2.325 tons. 82. How many lbs. of butter will a square box hold, its side being 6 inches and its depth 7 inches? Ans. 10 lbs. 2 oz. 83. What is the length of a pendulum which vibrates once in 7 seconds, in lat. 44? Ans. 159.7 feet, very nearly. 84. What is the ullage of a lying cask, whose capacity is 60 wine gallons, the bung diameter being 27 inches, and the depth of the liquor under the bung 10 inches? Ans. 20.2776 gallons, |