44. The angle subtended by a tower on an inclined plane is, at a certain point, 42° 17'; 325 ft. farther down, it is 21° 47'. The inclination of the plane is 8° 53'. Find the height of the tower. 45. A cape bears north by east, as seen from a ship. The ship sails northwest 30 miles, and then the cape bears east. How far is it from the second point of observation? 46. Two observers, stationed on opposite sides of a cloud, observe its angles of elevation to be 44° 56' and 36° 4'. Their distance from each other is 700 ft. What is the linear height of the cloud? 47. From a point B at the foot of a mountain, the elevation of the top A is 60°. After ascending the mountain one mile, at an inclination of 30° to the horizon, and reaching a point C, the angle ACB is found to be 135°. Find the height of the mountain in feet. 48. From a ship two rocks are seen in the same right line with the ship, bearing N. 15° E. After the ship has sailed northwest 5 miles, the first rock bears east, and the second northeast. Find the distance between the rocks. 49. From a window on a level with the bottom of a steeple the elevation of the steeple is 40°, and from a second window 18 ft. higher the elevation is 37° 30'. Find the height of the steeple. 50. To determine the distance between two inaccessible objects by observing angles at the extremities of a line of known length. 51. Wishing to determine the distance between a church A and a tower B, on the opposite side of a river, I measure a line CD along the river (C being nearly opposite 4), and observe the angles ACB, 58° 20'; ACD, 95° 20'; ADB, 53° 30'; BDC, 98° 45'. CD is 600 ft. What is the distance required? 52. Wishing to find the height of a summit A, I measure a horizontal base line CD, 440 yds. At C, the elevation of A is 37° 18′, and the horizontal angle between D and the summit is 76° 18'; at D, the horizontal angle between C and the summit is 67° 14'. Find the height. 53. A balloon is observed from two stations 3000 ft. apart. At the first station the horizontal angle of the balloon and the other station is 75° 25', and the elevation of the balloon is 18°. The horizontal angle of the first station and the balloon, measured at the second station, is 64° 30'. Find the height of the balloon. 54. Two forces, one of 410 pounds, and the other of 320 pounds, make an angle of 51° 37'. Find the intensity and the direction of their resultant. 55. An unknown force, combined with one of 128 pounds, produces a resultant of 200 pounds, and this resultant makes an angle of 18° 24' with the known force. Find the intensity and direction of the unknown force. 56. At two stations, the height of a kite subtends the same angle A. The angle which the line joining one station and the kite subtends at the other station is B; and the distance between the two stations is a. Show that the height of the kite is a sin A sec B. 57. Two towers on a horizontal plane are 120 ft. apart. A person standing successively at their bases observes that the angular elevation of one is double that of the other; but, when he is half-way between them, the elevations are complementary. Prove that the heights of the towers are 90 and 40 ft. 58. To find the distance of an inaccessible point C from either of two points A and B, having no instruments to measure angles. Prolong CA to a, and CB to b, and join AB, Ab, and Ba. Measure AB, 500; αA, 100; aB, 560; bB, 100; and Ab, 550. 59. Two inaccessible points A and B, are visible from D, but no other point can be found whence both are visible. Take some point C, whence A and D can be seen, and measure CD, 200 ft.; ADC, 89°; ACD, 50° 30′. Then take some point E, whence D and B are visible, and measure DE, 200; BDE, 54° 30'; BED, 88° 30'. At D measure ADB, 72° 30'. Compute the distance AB. 60. To compute the horizontal distance between two inaccessible points A and B, when no point can be found whence both can be seen. Take two points C and D, distant 200 yds., so that A can be seen from C, and B from D. From measure CF, 200 yds. to F, whence A can be seen; and from D measure DE, 200 yds. to E, whence B can be seen. Measure AFC, 83°; ACD, 53° 30'; ACF, 54° 31'; BDE, 54° 30′; BDC, 156° 25'; DEB, 88° 30'. 61. A column in the north temperate zone is east-southeast of an observer, and at noon the extremity of its shadow is northeast of him. The shadow is 80 ft. in length, and the elevation of the column, at the observer's station, is 45°. Find the height of the column. 62. From the top of a hill the angles of depression of two objects situated in the horizontal plane of the base of the hill are 45° and 30°; and the horizontal angle between the two objects is 30°. Show that the height of the hill is equal to the distance between the objects. 63. Wishing to know the breadth of a river from A to B, I take AC, 100 yds. in the prolongation of BA, and then take CD, 200 yds. at right angles to AC. The angle BDA is 37° 18' 30". Find AB. 64. The sum of the sides of a triangle is 100. The angle at A is double that of B, and the angle at B is double that at Determine the sides. C. 65. If sin2A+ 5 cos2A=3, find A. 66. If sin2A=m cos A―n, find cos A. 67. Given sin A=m sin B, and tan A=n tan B, find sin A and cos B. 68. If tan2A+4 sin2A=6, find A. 69. If sin A=sin 2 A, find A. 70. If tan 2 A=3 tan A, find A. 71. Prove that tan 50° + cot 50°2 sec 10°. 72. Given a regular polygon of n sides, and calling one of them a, find expressions for the radii of the inscribed and the circumscribed circles in terms of n and a. If P, H, D are the sides of a regular inscribed pentagon, hexagon, and decagon, prove P2=H2+D2. AREAS. 73. Obtain the formula for the area of a triangle, given two sides b, c, and the included angle A. 74. Obtain the formula for the area of a triangle, given two angles A and B, and included side c. 75. Obtain the formula for the area of a triangle, given the three sides. 76. If a is the side of an equilateral triangle, show that a2√3 its area is 4 77. Two consecutive sides of a rectangle are 52.25 ch. and 38.24 ch. Find its area. 78. Two sides of a parallelogram are 59.8 ch. and 37.05 ch., and the included angle is 72° 10'. Find the area. 79. Two sides of a parallelogram are 15.36 ch. and 11.46 ch., and the included angle is 47° 30'. Find its area. 80. Two sides of a triangle are 12.38 ch. and 6.78 ch., and the included angle is 46° 24'. Find the area. 81. Two sides of a triangle are 18.37 ch. and 13.44 ch., and they form a right angle. Find the area. 82. Two angles of a triangle are 76° 54′ and 57° 33′ 12′′, and the included side is 9 ch. Find the area. 83. Two sides of a triangle are 19.74 ch. and 17.34 ch. The first bears N. 82° 30′ W.; the second S. 24° 15' E. Find the area. 84. The three sides of a triangle are 49 ch., 50.25 ch., and 25.69 ch. Find the area. 85. The three sides of a triangle are 10.64 ch., 12.28 ch., and 9 ch. Find the area. 86. The sides of a triangular field, of which the area is 14 acres, are in the ratio of 3, 5, 7. Find the sides. 87. In the quadrilateral ABCD we have AB, 17.22 ch,; AD, 7.45 ch.; CD, 14.10 ch.; BC, 5.25 ch.; and the diagonal AC, 15.04 ch. Find the area. 88. The diagonals of a quadrilateral are a and b, and they intersect at an angle D. Show that the area of the quadrilateral is ab sin D. 89. The diagonals of a quadrilateral are 34 and 56, intersecting at an angle of 67°. Find the area. 90. The diagonals of a quadrilateral are 75 and 49, intersecting at an angle of 42°. Find the area. 91. Show that the area of a regular polygon of n sides, of na 2 180° 4 which one is a, is cot n 92. One side of a regular pentagon is 25. 93. One side of a regular hexagon is 32. Find the 'area. Find the area. |