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]
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Solution
Concept: 2 Marks Application: 2 Marks
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=xmetres. Then,
BD=BC=(x+60)m
From right ΔPQC, we have
PQCQ=cot30∘=√3
⇒PQxm=√3⇒PQ=x√3m
From right ΔPQD, we have
PQQD=cot60∘=1√3
⇒PQ(x+60+60)m=1√3⇒PQ=(x+120)√3m
Equating the values of PQ from (i) and (ii), we get
x√3=(x+120)√3m
⇒3x=x+120⇒2x=120⇒x=60
∴ height of the cloud from the surface of the lake
=BC=(60+x)m=(60+60)m=120m
Hence, the height of the cloud from the surface of the lake is 120 metres.