552 CHAPTER 8 Applications of Trigonometric Functions Solution Figure 14(b) shows two triangles that replicate Figure 14(a). The height of the statue of Ceres will be b b. ′ − To find b and b,′ refer to Figure 14(b). Figure 14 400 ft (a) 55.18 56.58 400 ft (b) 55.18 b 400 ft 56.58 b9 Now Work PROBLEM 71 ° = ° = ′ = ° ≈ ′ = ° ≈ b b b b tan55.1 400 tan56.5 400 400 tan 55.1 573.39 400tan 56.5 604.33 The height of the statue is approximately 604.33 573.39 30.94 feet 31 feet. − = ≈ The Gibb’s Hill Lighthouse, Southampton, Bermuda In operation since 1846, the Gibb’s Hill Lighthouse stands 117 feet high on a hill 245 feet high, so its beam of light is 362 feet above sea level. A brochure states that the light can be seen on the horizon about 26 miles distant. Verify the accuracy of this statement. Solution EXAMPLE 10 Figure 15 illustrates the situation. The beam of light hits the surface of Earth and forms a right angle with a radial line to the center of Earth, whose length is 3960 miles. The line from the top of the lighthouse to the center of Earth forms a triangle. The central angle ,θ positioned at the center of Earth, satisfies the equation cos 3960 3960 362 5280 0.999982687 θ = + ≈ 1 mile 5280 feet = Solving for θ yields cos 0.999982687 0.33715 20.23 1 θ ( ) ≈ ≈ ° ≈ − ′ The brochure does not indicate whether the distance is measured in nautical miles or statute miles. Let’s calculate both distances. The distance s in nautical miles (refer to Problem 120, p. 395) is the measure of the angle θ in minutes, so s 20.23 ≈ nautical miles. The distance s in statute miles is given by the formula s r ,θ = where θ is measured in radians. Then, since 0.33715 0.00588 radian θ ≈ ° ≈ ↑ 1 180 radian π ° = this means that s r 3960 0.00588 23.3 miles θ = ≈ ⋅ ≈ In either case, it would seem that the brochure overstated the distance somewhat. Figure 15 3960 mi 3960 mi 362 ft u s NOTE Since the radius is measured in miles, we must convert the height of the tower from feet to miles. j
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