Question

The angle of elevation of a cloud from a point 200 metres above the lake is 30° and the angle of depression of its reflection in the lake is 60°. Find the height of the cloud.

Answer 1

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^circ`

⇒ `(x - 200)/y = 1/sqrt(3)`

⇒ `y = sqrt(3) (x - 200)`   ...(i)

Also, in right-angled ΔC'AP,

⇒  `(C'A)/(PA) = tan 60^circ`

⇒ `(x + 200)/y = sqrt(3)`

⇒ `x + 200 = sqrt(3)y`

⇒ `y = ( x + 200)/sqrt(3)`   ...(ii)

From (i) and (ii),

⇒ `(x + 200)/sqrt(3) = sqrt(3) (x - 200)`

⇒ x + 200 = 3(x – 200)

⇒ x + 200 = 3x – 600

⇒ 2x = 800

⇒ x = 400

Hence, the height of the cloud = 400 m.