550 CHAPTER 8 Applications of Trigonometric Functions *An instrument used in surveying to measure angles. Finding the Width of a River A surveyor can measure the width of a river by setting up a transit* at a point C on one side of the river and taking a sighting of a point A on the other side. Refer to Figure 10. After turning through an angle of 90° at C, the surveyor walks a distance of 200 meters to point B. Using the transit at B, the angle θ is measured and found to be 20 .° What is the width of the river rounded to the nearest meter? Solution Now Work PROBLEM 49 Finding the Inclination of a Mountain Trail A straight trail leads from the Alpine Hotel, elevation 8000 feet, to a scenic overlook, elevation 11,100 feet. The length of the trail is 14,100 feet. What is the inclination (grade) of the trail? That is, what is the measure of angle B in Figure 11? Now Work PROBLEM 55 Vertical heights can sometimes be measured using either the angle of elevation or the angle of depression. If a person is looking up at an object, the acute angle measured from the horizontal to a line of sight to the object is called the angle of elevation. See Figure 12(a) on the next page. As seen in Figure 10, the width of the river is the length of side b, and a and θ are known. Use the facts that b is opposite θ and a is adjacent to θ and write b a tanθ = which leads to ° = = ° ≈ b b tan20 200 200 tan 20 72.79 meters The width of the river is 73 meters, rounded to the nearest meter. Solution From Figure 11, the length of the side opposite angle B is 11,100 8000 3100 − = feet, and the length of the hypotenuse is 14,100 feet. Angle B satisfies the equation B sin 3100 14,100 = Using a calculator, B sin 3100 14,100 12.7 1 = ≈ ° − The inclination (grade) of the trail is approximately 12.7 .° EXAMPLE 6 Figure 10 a 5 200 m C B A b u 5 20° EXAMPLE 7 Figure 11 Trail 14,100 ft Overlook elevation 11,100 ft , Hotel 3100 ft Elevation 8000 ft B
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