RIGHT TRIANGLES. 2. The angle of elevation of a tower is 48° 19' 14", and the distance of its base from the point of observation is 95 ft. Find the height of the tower, and the distance of its top from the point of observation. 3. From a mountain 1000 ft. high, the angle of depression of a ship is 77° 35' 11". Find the distance of the ship from the summit of the mountain. 4. A flag-staff 90 ft. high, on a horizontal plane, casts a shadow of 117 ft. Find the altitude of the sun. 5. When the moon is setting at any place, the angle at the moon subtended by the earth's radius passing through that place is 57'3". If the earth's radius is 3956.2 miles, what is the moon's distance from the earth's centre ? 6. The angle at the earth's centre subtended by the sun's radius is 16' 2", and the sun's distance is 92,400,000 miles. Find the sun's diameter in miles. 7. The latitude of Cambridge, Mass., is 42° 22' 49". What is the length of the radius of that parallel of latitude? 8. At what latitude is the circumference of the parallel of latitude half of that of the equator? 9. In a circle with a radius of 6.7 is inscribed a regular polygon of thirteen sides. Find the length of one of its sides. 10. A regular heptagon, one side of which is 5.73, is inscribed in a circle. Find the radius of the circle. 11. A tower 93.97 ft. high is situated on the bank of a river. The angle of depression of an object on the opposite bank is 25° 12' 54". Find the breadth of the river. 12. From a tower 58 ft. high the angles of depression of two objects situated in the same horizontal line with the base of the tower, and on the same side, are 30° 13′ 18′′ and 45° 46' 14". Find the distance between these two objects. 13. Standing directly in front of one corner of a flat-roofed house, which is 150 ft. in length, I observe that the horizontal angle which the length subtends has for its cosine √, and that the vertical angle subtended by its height has for its sine 3 √34 What is the height of the house? 14. A regular pyramid, with a square base, has a lateral edge 150 ft. in length, and the length of a side of its base is 200 ft. Find the inclination of the face of the pyramid to the base. 15. From one edge of a ditch 36 ft. wide, the angle of elevation of a wall on the opposite edge is 62° 39' 10". Find the length of a ladder which will reach from the point of observation to the top of the wall. 16. The top of a flag-staff has been broken off, and touches the ground at a distance of 15 ft. from the foot of the staff. The length of the broken part being 39 ft., find the whole length of the staff. 17. From a balloon, which is directly above one town, is observed the angle of depression of another town, 10° 14′ 9′′. The towns being 8 miles apart, find the height of the balloon. 18. From the top of a mountain 3 miles high the angle of depression of the most distant object which is visible on the earth's surface is found to be 2° 13' 50". Find the diameter of the earth. 19. A ladder 40 ft. long reaches a window 33 ft. high, on one side of a street. Being turned over upon its foot, it reaches another window. 21 ft. high, on the opposite side of the street. Find the width of the street. 20. The height of a house subtends a right angle at a window on the other side of the street; and the elevation of the top of the house, from the same point, is 60°. The street is 30 ft. wide. How high is the house? 21. A lighthouse 54 ft. high is situated on a rock. The elevation of the top of the lighthouse, as observed from a ship, is 4° 52', and the elevation of the top of the rock is 4° 2'. Find the height of the rock, and its distance from the ship. 22. A man in a balloon observes the angle of depression of an object on the ground, bearing south, to be 35° 30'; the balloon drifts 24 miles east at the same height, when the angle of depression of the same object is 23° 14'. Find the height of the balloon. 23. A man standing south of a tower, on the same horizontal plane, observes its elevation to be 54° 16'; he goes east 100 yds., and then finds its elevation is 50° 8'. Find the height of the tower. 24. The elevation of a tower at a place A south of it is 30°; and at a place B, west of A, and at a distance of a from it, the elevation is 18°. Show that the height of the tower is α √(2+2 √5) ; the tangent of 18° being √5-1 √(10+2 √√5) 25. A pole is fixed on the top of a mound, and the angles of elevation of the top and the bottom of the pole are 60° and 30° respectively. Prove that the length of the pole is twice the height of the mound. 26. At a distance (a) from the foot of a tower, the angle of elevation (A) of the top of the tower is the complement of the angle of elevation of a flag-staff on top of it. Show that the length of the staff is 2a cot 2 A. 27. A line of true level is a line every point of which is equally distant from the centre of the earth. A line drawn tangent to a line of true level at any point is a line of apparent level. If at any point both these lines are drawn, and extended one mile, find the distance they are then apart. 28. In Problem 2, determine the effect upon the computed height of the tower, of an error in either the angle of elevation or the measured distance. OBLIQUE TRIANGLES. 29. To determine the height of an inaccessible object situated on a horizontal plane, by observing its angles of elevation at two points in the same line with its base, and measuring the distance of these two points. 30. The angle of elevation of an inaccessible tower, situated on a horizontal plane, is 63° 26'; at a point 500 ft. farther from the base of the tower the elevation of its top is 32° 14′. Find the height of the tower. From the 31. A tower is situated on the bank of a river. opposite bank the angle of elevation of the tower is 60° 13', and from a point 40 ft. more distant the elevation is 50° 19′. Find the breadth of the river. 32. A ship sailing north sees two lighthouses 8 miles apart, in a line due west; after an hour's sailing, one lighthouse. bears S.W., and the other S.S.W. Find the ship's rate. 33. To determine the height of an accessible object situated on an inclined plane. 34. At a distance of 40 ft. from the foot of a tower on an inclined plane, the tower subtends an angle of 41° 19'; at a point 60 ft. farther away, the angle subtended by the tower is 23° 45'. Find the height of the tower. 35. A tower makes an angle of 113° 12' with the inclined plane on which it stands; and at a distance of 89 ft. from its base, measured down the plane, the angle subtended by the tower is 23° 27'. Find the height of the tower. 36. From the top of a house 42 ft. high, the angle of elevation of the top of a pole is 14° 13'; at the bottom of the house it is 23° 19'. Find the height of the pole. 37. The sides of a triangle are 17, 21, 28; prove that the length of a line bisecting the greatest side and drawn from the opposite angle is 13. 38. A privateer, 10 miles S.W. of a harbor, sees a ship sail from it in a direction S. 80° E., at a rate of 9 miles an hour. In what direction, and at what rate, must the privateer sail in order to come up with the ship in 12 hours? 39. A person goes 70 yds. up a slope of 1 in 3 from the edge of a river, and observes the angle of depression of an object on the opposite shore to be 21°. Find the breadth of the river. 40. The length of a lake subtends, at a certain point, an angle of 46° 24', and the distances from this point to the two extremities of the lake are 346 and 290 ft. Find the length of the lake. 41. Two ships are a mile apart. The angular distance of the first ship from a fort on shore, as observed from the second ship, is 35° 14' 10"; the angular distance of the second ship from the fort, observed from the first ship, is 42° 11′ 53′′. Find the distance in feet from each ship to the fort. 42. Along the bank of a river is drawn a base line of 500 feet. The angular distance of one end of this line from an object on the opposite side of the river, as observed from the other end of the line, is 53°; that of the second extremity from the same object, observed at the first, is 79° 12'. Find the perpendicular breadth of the river. 43. A vertical tower stands on a declivity inclined 15° to the horizon. A man ascends the declivity 80 ft. from the base of the tower, and finds the angle then subtended by the tower to be 30°. Find the height of the tower. |