Question

# The angle of elevation of a cloud from a point 60 m above a lake is 30∘ and the angle of depression of the reflection of cloud in the lake is 60∘. Find the height of the cloud.  [4 MARKS]

Solution

## Let AB be the surface of the lake and let P be a point vertically above A such that AP = 60 m Let C be the position of the cloud and let D be its reflection in the lake. Draw PQ⊥CD, Then, ∠QPC=30∘,∠QPD=60∘, BQ = AP = 60 m Let CQ=x metres. Then, BD=BC=(x+60) m From right ΔPQC, we have PQCQ=cot30∘=√3 ⇒PQx m=√3⇒PQ=x√3 m  -----------(i) From right ΔPQD, we have PQQD=cot60∘=1√3 ⇒PQ(x+60+60)m=1√3⇒PQ=(x+120)√3 m  -------(ii) Equating the values of PQ from (i) and (ii), we get x√3=(x+120)√3 m ⇒3x=x+120⇒2x=120⇒x=60 ∴ height of the cloud from the surface of the lake =BC=(60+x)m=(60+60) m=120 m Hence, the height of the cloud from the surface of the lake is 120 metres. Alernative Method,                                                                                                                          [12 MARK] In ΔABE tan 30=hx 1√3=hx⇒x=h√3……(1)                            [1 MARK] In ΔBDE tan 60=120+hx √3=120+hh√3 3h=120+h 2h=120 h=60 m                                                              [1 MARK] Hence, height of cloud above the lake = 60 + h                                                                = 60 + 60                                                                = 120 m            [12 MARK]

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