SECTION 8.2 The Law of Sines 567 41. Finding the Height of an Airplane An aircraft is spotted by two observers who are 1000 feet apart. As the airplane passes over the line joining them, each observer takes a sighting of the angle of elevation to the plane, as indicated in the figure. How high is the airplane? 258 1000 ft P Q 508 42. Finding the Height of the Bridge over the Royal Gorge The highest bridge in the United States is the bridge over the Royal Gorge of the Arkansas River in Colorado. Sightings to the same point at water level directly under the bridge are taken from each side of the 880-foot-long bridge, as indicated in the figure. How high is the bridge? Source: highestbridges.com 43. Land Dimensions A triangular plot of land has one side along a straight road measuring 200 feet. A second side makes a ° 50 angle with the road, and the third side makes a ° 43 angle with the road. How long are the other two sides? 44. Distance between Runners Two runners in a marathon determine that the angles of elevation of a news helicopter covering the race are ° 38 and ° 45 . If the helicopter is 1700 feet directly above the finish line, how far apart are the runners? 45. Landscaping Pat needs to determine the height of a tree before cutting it down to be sure that it will not fall on a nearby fence. The angle of elevation of the tree from one position on a flat path from the tree is ° 30 , and from a second position 40 feet farther along this path it is ° 20 . What is the height of the tree? 46. Construction A loading ramp 10 feet long that makes an angle of ° 18 with the horizontal is to be replaced by one that makes an angle of ° 12 with the horizontal. How long is the new ramp? 47. Commercial Navigation Naomi must fly home to St. Louis from a business meeting in Oklahoma City. One flight option flies directly to St. Louis, a distance of about 461.1 miles. A second flight option flies first to Kansas City and then connects to St. Louis. The bearing from Oklahoma City to Kansas City is ° N29.6 E, and the bearing from Oklahoma City to St. Louis is ° N57.7 E. The bearing from St. Louis to Oklahoma City is ° S57.7 W, and the bearing from St. Louis to Kansas City is ° N79.4 W. How many more frequent flyer miles will Naomi receive if she takes the connecting flight rather than the direct flight? Source: www.landings.com 48. Time Lost to a Navigation Error In attempting to fly from city P to city Q, an aircraft followed a course that was ° 10 in error, as indicated in the figure. After flying a distance of 50 miles, the pilot corrected the course by turning at point R and flying 300 miles farther. If the constant speed of the aircraft was 250 miles per hour, how much time was lost due to the error? 108 50 mi 300 mi R P Q 49. Rescue at Sea Coast Guard Station Able is located 150 miles due south of Station Baker. A ship at sea sends an SOS call that is received by each station. The call to Station Able indicates the bearing of the ship is ° N55 E; the call to Station Baker indicates the bearing of the ship is ° S60 E. (a) How far is each station from the ship? (b) If a helicopter capable of flying 200 miles per hour is dispatched from the station nearest the ship, how long will it take to reach the ship? 50. Distance to the Moon At exactly the same time, Tom and Alice measured the angle of elevation to the moon while standing exactly 300 km apart. The angle of elevation to the moon for Tom was ° 49.8974 , and the angle of elevation to the moon for Alice was ° 49.9312 . See the figure. To the nearest 1000 km, how far was the moon from Earth when the measurements were obtained? 51. Finding the Lean of the Leaning Tower of Pisa The famous Leaning Tower of Pisa was originally 184.5 feet high.* At a distance of 123 feet from the base of the tower, the angle of elevation to the top of the tower is found to be ° 60 . Find ∠RPQ indicated in the figure. Also, find the perpendicular distance from R to PQ. *On February 27, 1964, the government of Italy requested aid in preventing the tower from toppling. A multinational task force of engineers, mathematicians, and historians was assigned and met on the Azores islands to discuss stabilization methods.After over two decades of work on the subject, the tower was closed to the public in January 1990. During the time that the tower was closed, the bells were removed to relieve it of some weight, and cables were cinched around the third level and anchored several hundred meters away. Apartments and houses in the path of the tower were vacated for safety concerns. After a decade of corrective reconstruction and stabilization efforts, the tower was reopened to the public on December 15, 2001. Many methods were proposed to stabilize the tower, including the addition of 800 metric tons of lead counterweights to the raised end of the base. The final solution was to remove 38 cubic meters of soil from underneath the raised end. The tower has been declared stable for at least another 300 years. Source: https://www.history.com/this-day-in-history/leaning-tower-needs-help 880 ft 69.28 65.58 h 49.93128 Moon Tom 300 km Alice 49.89748 R P Q 608 123 ft 184.5 ft
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