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The angle of elevation of a cloud from a point h metres above a lake is α and the angle of depression of the reflection of the cloud in the lake is . Prove that the height of the cloud is metres. OR From a window, h metres high above the ground, of a house in a street, the angles of elevation and depression of the top and the foot of another house on the opposite side of the street are α and respectively. Show that the height of the opposite house is h(1+tanα cot) metres.
Asked by Topperlearning User | 02 Dec, 2013, 10:33: AM

Let C be the cloud and C' be its reflection. Let the height of the cloud be H metres.

BC=BC'=H m

Now BM=AP= h m, therefore, CM= H-h and MC' = H+h

In CPM,

= tan

… (i)

In PMC',

… (ii)

From (i) and (ii),

Htan - htan = Htan+htan

Htan - Htan = htan + htan

H(tan - tan) = h(tan+tan)

Hence, the height of the cloud is metres.

OR

Let B be the window of a house AB and let CD be the other house. Then, AB = EC = h metres.

Let CD = H metres. Then, ED= (H-h) m.

In BED,

cot =

BE = (H-h) cotα … (i)

In ACB,

cot =

AC=h.cot … (ii)

But BE=AC

(H-h) cot = hcot

H=

H = h(1+tan cot)

Thus, the height of the opposite house is h(1+tan cot) metres.

Answered by | 02 Dec, 2013, 12:33: PM

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