A man at M, standing 100 m away from the base (P) of a chimney of height 50 m, observes the angle of elevation of the highest point (Q) of the smoke to be 45°. The highest point of the chimney is at R. Further P, R and Q are in a straight line and the straight line is perpendicular to PM. What is the angle RMQ equal to?
B. ^ -1 ( 1 3 )
Let PM be the distance of the man from the base of the chimney, so PM = 100 m. Let PR be the height of the chimney, so PR = 50 m. Let Q be the highest point of the smoke. The angle of elevation of Q from M is PMQ = 45^ . In PMQ, ( PMQ) = PQ PM (45^ ) = PQ 100 PQ = 100 m. In PMR, ( PMR) = PR PM = 50 100 = 1 2 . The required angle is RMQ = PMQ - PMR = 45^ - PMR. Taking tangent on both sides: ( RMQ) = (45^ - PMR) = (45^ ) - ( PMR) 1 + (45^ ) ( PMR) ( RMQ) = 1 - 1 2 1 + 1 1 2 = 1 2 3 2 = 1 3 RMQ = ^ -1 ( 1 3 ) Answer: ^ -1 ( 1 3 )
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