Q17 of 35 Page 219

If the angle of elevation of a cloud from a point h metres above a lake has measure a and the angle of depression of its reflection in the lake has measure β, prove that the height of the h(tani3 + tana) cloud is .

Let AB be the surface of the lake. E is the point above h metre from A.



AE = h


Let height of the cloud BD = l


Let F be the reflection of kite C.


Horizontal line EC intersects BD in C.


BF = l


∠DEC = α, ∠CEF = ß


Here, AE = BC = h


CD = BD – BC = l – h and


CF = BF + BC = l + h


In ∆ECD,



……(1)


In ∆ECF,



…….(2)


From results (1) and (2),




Using componendo – dividendo,






The height of the cloud from the surface of the lake is m.


Hence proved.


More from this chapter

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15

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A man on the top of a vertical tower observes a car moving at a uniform speed coming directly towards it. If it takes 12 minutes for the angle of depression to change its measure from 30 to 45, how soon after this, will the car reach the tower?

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From the top of a building , 60 m high, the angles of depression of the top and bottom at a vertical lamp post are observed to have measure 30 and 60 respectively. Find,

(1) the horizontal distance between building and lamp post.


(2) the height of the lamp post.


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