564 CHAPTER 8 Applications of Trigonometric Functions Figure 31 (a) 358 (b) 2 m 900 m h c b A C 358 478 2 m 900 m 478 h FUN FACT This way of measuring altitude is still used when radio receivers cannot be placed at the top of a mountain. Surveyors measure the angle of elevation with the assistance of a glorified, highly accurate protractor, a telescopic device known as a theodolite. j Solution Figure 31(b) shows the triangles that model the situation in Figure 31(a). Since + ° = ° C 47 180 , this means that = ° C 133 . Also, since + + ° = ° A C 35 180 , this means that = °−°−= °− °=° A C 180 35 145 133 12 . Use the Law of Sines to find c. = = ° ° ≈ A a C c c sin sin 900 sin133 sin12 3165.86 Using the larger right triangle gives ° = = ° ≈ ≈ b c b sin35 3165.86 sin35 1815.86 1816 meters The height of the peak from ground level is approximately + = 1816 2 1818 meters. = ° = ° = A C a 12 , 133 , 900 Now Work PROBLEM 39 Rescue at Sea Coast Guard Station Zulu is located 120 miles due west of Station X-ray.A ship at sea sends an SOS call that is received by each station. The call to Station Zulu indicates that the bearing of the ship from Zulu is ° N40 E ( ° 40 east of north). The call to Station X-ray indicates that the bearing of the ship from X-ray is ° N30 W ( ° 30 west of north). (a) How far is each station from the ship? (b) If a helicopter capable of flying 200 miles per hour is dispatched from the nearest station to the ship, how long will it take to reach the ship? EXAMPLE 7 Figure 32 Zulu N S W E X-ray b a C 608 308 508 408 120 mi Solution (a) Figure 32 illustrates the situation. The angle C is found to be = °− °− ° = ° C 180 50 60 70 The Law of Sines can now be used to find the two distances a and b that are needed. ° = ° = ° ° ≈ ° = ° = ° ° ≈ a a b b sin 50 sin 70 120 120 sin50 sin70 97.82 miles sin 60 sin 70 120 120 sin60 sin70 110.59 miles Station Zulu is about 111 miles from the ship, and Station X-ray is about 98 miles from the ship.
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