SOLUTION: A tower TR and an observer at O. |OR| = 84 m and the angle of elevation of the top of the tower T from O is 57°. (a) Calculate, correct to three significant figures, the height o

Question 1210335: A tower TR and an observer at O. |OR| = 84 m and the angle of elevation of the top of the tower T from O is 57°.
(a) Calculate, correct to three significant figures, the height of the tower.
(b) The observer at O, moved away from the tower in the same straight line until the angle of elevation of T is 49°. Find, correct to two decimal places, how far the observer moved backwards.
Answer by ikleyn(52780) (Show Source): You can put this solution on YOUR website!
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A tower TR and an observer at O. |OR| = 84 m and the angle of elevation of the top of the tower T from O is 57°.
(a) Calculate, correct to three significant figures, the height of the tower.
(b) The observer at O, moved away from the tower in the same straight line until the angle of elevation of T is 49°.
Find, correct to two decimal places, how far the observer moved backwards.
~~~~~~~~~~~~~~~~~~~~~~~~~~

In this problem,  'T'  denotes the top of the tower;  'R'  denotes its base.



(a)  In part (a), we have a right-angled triangle with vertical leg |TR| and horizontal leg |OR| = 84 m.

     Recall the definition of tangent of an acute angle in a right-angled triangle

                     opposite leg
          = ---------------.
                     adjacent leg


     In this part, the angle is 57°, the opposite leg is the unknown height of the tower h = |TR|,
     the adjacent leg is the given horizontal distance from the observer to the base of the tower  |OR| = 84 m.


     So, we write

          tan(57°) = .


     From this equation,  the height of the tower  'h'  is

          h = |OR|*tan(57°) = 84*1.53986496381 = 129.35 m  (rounded).


     Thus the height of the tower is 129.35 m.    ANSWER



(b)  In part (b), we have a right-angled triangle with vertical leg |TR| = h = 129.35 m and horizontal leg x,
     where x is the new horizontal distance from the observer to the base of the tower.

     Using the definition of the tangent of an acute angle in a right-angled triangle, we can write

          tan(49°) = .


     From this equation, the new distance from the tower to the observer is

          x =  =  = 112.44 m  (rounded).


     It means that the observer moved backward  112.44 - 84 =  28.44 meters.    ANSWER

Solved.

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