568 CHAPTER 8 Applications of Trigonometric Functions 56. Finding Distances A forest ranger is walking on a path inclined at °5 to the horizontal directly toward a 100-foot-tall fire observation tower. The angle of elevation from the path to the top of the tower is ° 40 . How far is the ranger from the tower at this time? 100 ft Path Horizontal 408 58 57. Great Pyramid of Cheops One of the original Seven Wonders of the World, the Great Pyramid of Cheops was built about 2580 bc. Its original height was 480 feet 11 inches, but owing to the loss of its topmost stones, it is now shorter. Find the current height of the Great Pyramid using the information given in the figure. Source: Guinness Book of World Records. 40.38 46.278 200 ft 100 ft 58. Determining the Height of an Aircraft Two sensors are spaced 700 feet apart along the approach to a small airport. When an aircraft is nearing the airport, the angle of elevation from the first sensor to the aircraft is ° 20 , and from the second sensor to the aircraft it is ° 15 . Determine how high the aircraft is at this time. 59. Mercury The distance from the Sun to Earth is approximately ( ) 149,600,000 kilometers km . The distance from the Sun to Mercury is approximately 57,910,000 km. The elongation angle α is the angle formed between the line of sight from Earth to the Sun and the line of sight from Earth to Mercury. See the figure. Suppose that the elongation angle for Mercury is ° 15 . Use this information to find the possible distances between Earth and Mercury. Mercury Sun Mercury Earth a 52. Crankshafts on Cars On a certain automobile, the crankshaft is 3 inches long and the connecting rod is 9 inches long (see the figure). At the time when ∠OPQ is ° 15 , how far is the piston ( )P from the center ( )O of the crankshaft? Q P 9 in. O 3 in. 158 53. Constructing a Highway U.S. 41, a highway whose primary direction is north– south, is being constructed along the west coast of Florida. Near Naples, a bay obstructs the straight path of the road. Since the cost of a bridge is prohibitive, engineers decide to go around the bay. The figure shows the path that they decide on and the measurements taken. What is the length of highway needed to go around the bay? 54. Calculating Distances at Sea The navigator of a ship at sea spots two lighthouses that she knows to be 3 miles apart along a straight seashore. She determines that the angles formed between two line-of-sight observations of the lighthouses and the line from the ship directly to shore are ° 15 and ° 35 . See the figure. (a) How far is the ship from lighthouse P? (b) How far is the ship from lighthouse Q? (c) How far is the ship from shore? Ocean 358 3 mi 158 P Q 55. Designing an Awning An awning that covers a sliding glass door that is 88 inches tall forms an angle of ° 50 with the wall. The purpose of the awning is to prevent sunlight from entering the house when the angle of elevation of the Sun is more than ° 65 . See the figure. Find the length L of the awning. 2 mi 1408 1358 Ocean Pelican Bay 41 1 – 8 mi 1 – 8 mi Clam Bay Highway U.S. 41 658 Step 880 508 L
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