Q. 114.6( 5 Votes )


The angle of elevation of a cliff from a fixed point A is θ. After going up adistance of k metres towards the top of the cliff at an angle φ, itis found that the angle of elevation is α. Show that the height ofthe cliff in metre is https://gs-post-images.grdp.co/user_files/63094/15047/images/extra-9_files/11d20.gif .

Answer :


https://gs-post-images.grdp.co/user_files/63094/15047/images/extra-9_files/Image1109.gif

Given:

Angle of elevation from fixed point A = θ

Distance covered = k m

Angle after the distance k metres is covered = ϕ

In right angled Δ FED,


FE = k m


sin φ =
ED / FE

ED = FE sin φ

= k sin φ


Again, cos φ = FD / FE

FD = FE cos φ

= k cos f .....................(1)


In right angled Δ AFC,

FC / AC = cot θ


FC = AC cot q................................... (2)


FC - FD = DC


But, DC = EB

EB = FC – FD


Thus, EB = AC cot θ - k cos f …………………. (from (1) and (2))


Now, AB = AC - BC


But, BC = ED = k sin φ

AB = AC - k sin φ


In right angled Δ AEB, cot α = EB / AB


AC cot θ - k cos φ = AC cot α - k sin φ cot α


AC cot θ - AC cosα = k cos φ - k sin φ cot α


AC (cot θ - cot α ) = k cos φ - k sin φ cotα


Hence proved the height of the cliff in metres is

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