= b=149, A 70°42'30", B=39° 18′ 28". 15540. c=8.025, B=100° 5′23′′, C=31° 6'12". 46.177. a=242 yards, b=1212 yards, c=1450 yards. Ans. 6 acres. 87. 66 a=7.152, b=8.263, c=9.375. 28.47717. 88. The sides of a triangle are as 2:3:4: show that the radii of the escribed circles are as :: 1. 89. The area of a triangle is an acre; two of its sides are 127 yards and 150 yards: find the angle between them. Ans. 30° 32' 23'. 90. The adjacent sides of a parallelogram are 5 and 8, and they include an angle of 60°: find (1) the two diagonals, and (2) the area. Ans. (1) 7, √129; (2) 20 √3. and 91. Two angles of a triangular field are 224° and 45°, the length of the side opposite the latter is a furlong. Show that the field contains 2 acres. HEIGHTS AND DISTANCES. 92. At a point 200 feet in a horizontal line from the foot of a tower, the angle of elevation of the top of the tower is observed to be 60°: find the height of the tower. Ans. 346 feet. 93. From the top of a vertical cliff, the angle of depression of a point on the shore 150 feet from the base of the cliff, is observed to be 30°: find the height of the cliff. Ans. 86.6 feet. 94. From the top of a tower 117 feet high, the angle of depression of the top of a house 37 feet high is observed to be 30°: how far is the top of the house from the tower? Ans. 138.5 feet. 95. The shadow of a tower in the sunlight is observed to be 100 feet long, and at the same time the shadow of a lamp-post 9 feet high is observed to be 3√3 feet long: find the angle of elevation of the sun, and height of the tower. Ans. 60°; 173.2 feet. 96. A flagstaff 25 feet high stands on the top of a house; from a point on the plain on which the house stands, the angles of elevation of the top and bottom of the flagstaff are observed to be 60° and 45° respectively: find the height of the house above the point of observation. Ans. 34.15 feet. 97. From the top of a cliff 100 feet high, the angles of depression of two ships at sea are observed to be 45° and 30° respectively; if the line joining the ships points directly to the foot of the cliff, find the distance between the ships. Ans. 73.2. 98. A tower 100 feet high stands on the top of a cliff; from a point on the sand at the foot of the cliff the angles of elevation of the top and bottom of the tower are observed to be 75° and 60° respectively: find the height of the cliff. Ans. 86.6 feet. 99. A man walking a straight road observes at one milestone a house in a direction making an angle of 30° with the road, and at the next milestone the angle is 60°: how far is the house from the road? Ans. 1524 yds. 100. A man stands at a point A on the bank AB of a straight river and observes that the line joining A to a post C on the opposite bank makes with AB an angle of 30°. He then goes 400 yards along the bank to B and finds that BC makes with BA an angle of 60°: find the breadth of the river. Ans. 173.2 yards. 101. From the top of a hill the angles of depression of the top and bottom of a flagstaff 25 feet high at the foot of the hill are observed to be 45° 13′ and 47° 12' respectively: find the height of the hill. Ans. 373 feet. 102. From each of two stations, east and west of each other, the altitude of a balloon is observed to be 45°, and its bearings to be respectively N.W. and N.E.; if the stations be 1 mile apart, find the height of the balloon. Ans. 3733 feet. 103. The angle of elevation of a balloon from a station due south of it is 60°, and from another station due west of the former and distant a mile from it is 45°: find the height of the balloon. Ans. 6468 feet. 104. Find the height of a hill, the angle of elevation at its foot being 60°, and at a point 500 yards from the foot along a horizontal plane 30°. Ans. 250√3 yards. 105. A tower 51 feet high has a mark at a height of 25 feet from the ground: find at what distance from the foot the two parts subtend equal angles. 106. The angles of a triangle are as 1: 2: 3, and the perpendicular from the greatest angle on the opposite side is 30 yards find the sides. Ans. 20√3, 60, 40√3. : 107. At two points A, B, an object DE, situated in the same vertical line CE, subtends the same angle a; if AC, BC be in the same right line, and equal to a and b, respectively, prove DE = (a+b) tan α. 108. From a station B at the foot of an inclined plane BC the angle of elevation of the summit A of a mountain is 60°, the inclination of BC is 30°, the angle BCA 135°, and the length of BC is 1000 yards: find the height of A over B. Ans. 500 (3+√3) yards. 109. A right triangle rests on its hypotenuse, the length of which is 100 feet; one of the angles is 36°, and the inclination of the plane of the triangle to the horizon is 60°: find the height of the vertex above the ground. Ans. 25√3 cos 18°. 110. A station at A is due west of a railway train at B; after traveling N. W. 6 miles, the bearing of A from the train is S. 221° W.: find the distance AB. Ans. 6 miles. 111. The angles of depression of the top and bottom of a column observed from a tower 108 feet high are 30° and 60° respectively: find the height of the column. Ans. 72 feet. 112. At the foot of a mountain the elevation of its summit is found to be 45°. After ascending for one mile, at a slope of 15°, towards the summit, its elevation is found to be 60°: find the height of the mountain. 113. A and B are two stations on a hillside. The inclination of the hill to the horizon is 30°. The distance between A and B is 500 yards. C is the summit of another hill in the same vertical plane as A and B, on a level with A, but at B its elevation above the horizon is 15°: find the distance between A and C. Ans. 500(√3+1). 114. From the top of a cliff the angles of depression of the top and bottom of a lighthouse 97.25 feet high are observed to be 23° 17' and 24° 19' respectively how much higher is the cliff than the lighthouse? Ans. 1942 feet. 115. The angle of elevation of a balloon from a station due south of it is 47° 18′ 30", and from another station due west of the former, and distant 671.38 feet from it, the elevation is 41° 14': find the height of the balloon. Ans. 1000 feet. 116. A person standing on the bank of a river observes the elevation of the top of a tree on the opposite bank to be 51°; and when he retires 30 feet from the river's bank he observes the elevation to be 46°: find the breadth of the river. Ans. 155.823 feet. 117. From the top of a hill I observe two milestones on the level ground in a straight line before me, and I find their angles of depression to be respectively 5° and 15°: find the height of the hill. Ans. 228.6307 yards. 118. A tower is situated on the top of a hill whose angle of inclination to the horizon is 30°. The angle subtended by the tower at the foot of the hill is found by an observer to be 15°; and on ascending 485 feet up the hill the tower is found to subtend an angle of 30°: find (1) the height of the tower, and (2) the distance of its base from the foot of the hill. Ans. (1) 280.015; (2) 765.015 feet. 119. The angle of elevation of a tower at a place A due south of it is 30°; and at a place B, due west of A, and at a distance a from it, the elevation is 18°: show that the height of the tower is α √2+2 √5 |