16. At the foot of a mountain the elevation of its summit is 45°. After ascending 1000 m towards the mountain up a slope of 30° inclination, the elevation is found to be 60° Find the height of the mountain. (3)
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height of mountain = OA =H
In ΔAOB,
tan (ABO) = AO/OB
⇒ tan 45 = AO/OB
⇒ 1 = AO/OB
⇒ OB = AO = H
In ΔPBR,
sin 30 = PR/PB
⇒ 1/2 = y/1000
⇒ y = 1000×(1/2) = 500m
Thus, AQ = AO - OQ = (H-500)m
cos 30 = RB/PB
⇒ √3 / 2 = x/1000
⇒ BR = 1000×(√3 / 2)
⇒ BR = 500√3m
thus OR = BO - BR = (H - 500√3)m
thus PQ = (H-500√3)m (since OP=PQ)
In ΔAPQ,
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