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Question
The angle of elevation of a cloud from a point 200 metres above a lake is 30° and the angle of depression of its reflection in the lake is 60°. Find the height of the cloud.
Solution
Let P be the point of observation and C, the position of cloud. CN ⊥ from C on the surface of the lake and C' be the reflection of the cloud in the lake so that
CN = NC' = x (say)
Then, PM = 200 m
∴ AN = MP = 200 m
CA = CN - AN = ( x - 200 ) m
C'A = NC' + AN = ( x + 200 ) m
Let, PA = y m
Then in right angled ΔPAC,
⇒ `(CA)/(PA) = tan 30°`
⇒ `(x - 200)/y = 1/sqrt3`
⇒ y = √3( x - 200) ....(i)
Also, in right angled ΔC'AP,
⇒ `(C'A)/(PA) = tan 60°`
⇒ `(x + 200)/y = sqrt3`
⇒ x + 200 = √3y
⇒ y = `( x + 200)/sqrt3` .....(ii)
From (i) and (ii),
⇒ `(x + 200)/sqrt3 = sqrt3(x - 200)`
⇒ x + 200 = 3( x - 200)
⇒ x + 200 = 3x - 600
⇒ 2x = 800
⇒ x = 400
Hence, the height of the cloud = 400 m.
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