594 CHAPTER 8 Applications of Trigonometric Functions In Problems 8–20, find the remaining angle(s) and side(s) of each triangle, if it (they) exists. If no triangle exists, say “No triangle.” 8. = ° = ° = A B a 50 , 30 , 1 9. = ° = = A a c 100 , 5, 2 10. = = = ° a c C 3, 1, 110 11. = = = ° a c B 3, 1, 100 12. = = = ° a b B 3, 5, 80 13. = = = a b c 2, 3, 1 14. = = = a b c 10, 7, 8 15. = = = ° a b C 1, 3, 40 16. = = = ° a b A 5, 3, 80 17. = = = a b c 1, 1 2 , 4 3 18. = = ° = a A b 3, 10 , 4 19. = = ° = ° a A B 4, 20 , 100 20. = = = ° c b A 5, 4, 70 In Problems 21–25, find the area of each triangle. 21. = = = ° a b C 2, 3, 40 22. = = = ° b c A 4, 10, 70 23. = = = a b c 4, 3, 5 24. = = = a b c 4, 2, 5 25. = ° = ° = A B a 50 , 30 , 1 26. Area of a Segment Find the area of the segment of a circle whose radius is 6 inches formed by a central angle of ° 50 . 27. Geometry The hypotenuse of a right triangle is 12 feet. If one leg is 8 feet, find the degree measure of each angle. 28. Finding the Width of a River Find the distance from A to C across the river illustrated in the figure. 31. Finding the Grade of a Mountain Trail A straight trail with a uniform inclination leads from a hotel, elevation 5000 feet, to a lake in a valley, elevation 4100 feet.The length of the trail is 4100 feet.What is the inclination (grade) of the trail? 32. Finding the Height of a Helicopter Two observers simultaneously measure the angle of elevation of a helicopter. One angle is measured as ° 25 , the other as ° 40 (see the figure). If the observers are 100 feet apart and the helicopter lies over the line joining them, how high is the helicopter? 50 ft C B A 25° 29. Finding the Distance to Shore The Willis Tower in Chicago is 1454 feet tall and is situated about 1 mile inland from the shore of Lake Michigan, as indicated in the figure.An observer in a pleasure boat on the lake directly in front of the Willis Tower looks at the top of the tower and measures the angle of elevation as °5 . How far offshore is the boat? Lake Michigan 5° 1 mi 1454 ft 30. Finding the Speed of a Glider From a glider 200 feet above the ground, two sightings of a stationary object directly in front are taken 1 minute apart (see the figure, top right). What is the speed of the glider? 100 ft 408 258 33. Constructing a Highway A highway whose primary directions are 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? Gulf Clam Bay 1 – 4 mi 1 – 4 mi C B 1208 1158 3 mi 200 ft 40° 10°
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