578 CHAPTER 5 Trigonometric Functions 54. Cloud Ceiling The U.S. Weather Bureau defines a cloud ceiling as the altitude of the lowest clouds that cover more than half the sky. To determine a cloud ceiling, a powerful searchlight projects a circle of light vertically on the bottom of the cloud. An observer sights the circle of light in the crosshairs of a tube called a clinometer. A pendant hanging vertically from the tube and resting on a protractor gives the angle of elevation. Find the cloud ceiling if the searchlight is located 1000 ft from the observer and the angle of elevation is 30.0° as measured with a clinometer at eyeheight 6 ft. (Assume three significant digits.) 30.08 6 ft 1000 ft Cloud Observer Searchlight 55. Height of Mt. Everest The highest mountain peak in the world is Mt. Everest, located in the Himalayas. The height of this enormous mountain was determined in 1856 by surveyors using trigonometry long before it was first climbed in 1953. This difficult measurement had to be done from a great distance. At an altitude of 14,545 ft on a different mountain, the straight-line distance to the peak of Mt. Everest is 27.0134 mi and its angle of elevation is u = 5.82°. (Data from Dunham, W., The Mathematical Universe, John Wiley and Sons.) (a) Approximate the height (in feet) of Mt. Everest. (b) In the actual measurement, Mt. Everest was over 100 mi away and the curvature of Earth had to be taken into account. Would the curvature of Earth make the peak appear taller or shorter than it actually is? 56. Error in Measurement A degree may seem like a very small unit, but an error of one degree in measuring an angle may be significant. For example, suppose a laser beam directed toward the visible center of the moon misses its assigned target by 30.0″. How far is it (in miles) from its assigned target? Take the distance from the surface of Earth to that of the moon to be 234,000 mi. (Data from A Sourcebook of Applications of School Mathematics by Donald Bushaw et al.) Concept Check The two methods of expressing bearing can be interpreted using a rectangular coordinate system. Suppose that an observer for a radar station is located at the origin of a coordinate system. Find the bearing of an airplane located at each point. Express the bearing using both methods. 57. 1-4, 02 58. 15, 02 59. 10, 42 60. 10, -22 61. 1-5, 52 62. 1-3, -32 63. 12, -22 64. 12, 22 Solve each problem. See Examples 5 and 6. 65. Distance Flown by a Plane A plane flies 1.3 hr at 110 mph on a bearing of 38°. It then turns and flies 1.5 hr at the same speed on a bearing of 128°. How far is the plane from its starting point? 14,545 ft 27.0134 mi u 128° x 38° N N
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