The angle of elevation of a cliff from a fixed point is - After going up a distance of k metres towards the top of the cliff at an angle of - it is found that the angle of elevation is - Show that the height of the cliff is kcos - - sin - cot -cot - - cot - metres-

Height of the cliff = AB In triangle EFC, EC=k = gt; EF=ksinϕ, #160; CF=kcosϕ=BD In triangle ABC, Let AB=x = gt; AC=xcosecθ ; BC=xco ...

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Standard X

Mathematics

Calculating Heights and Distances

The angle of ...

Question

The angle of elevation of a cliff from a fixed point is $θ$ . After going up a distance of k metres towards the top of the cliff at an angle of $ϕ$ , it is found that the angle of elevation is $α$ . Show that the height of the cliff is
$\frac{k (c o s ϕ - s i n ϕ c o t α)}{c o t θ - c o t α}$ metres.

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Solution

Height of the cliff $= A B$

In triangle EFC,
$E C = k$
=> $E F = k s i n ϕ, C F = k c o s ϕ = B D$

In triangle ABC, Let $A B = x$
=> $A C = x c o s e c θ; B C = x c o t θ$

$F B = B C - C F = x c o t θ - k c o s ϕ = E D$
$A D = E D t a n α = t a n α (x c o t θ - k c o s ϕ)$
$A D = A B - B D = x - k s i n ϕ$
=> $t a n α = \frac{x - k s i n α}{x c o t θ - k c o s ϕ}$

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The angle of elevation of a cliff from a fixed point is θ. After going up a distance of k metres towards the top of the cliff at an angle of ϕ, it is found that the angle of elevation is α. Show that the height of the cliff isk(cosϕ−sinϕcotα)cotθ−cotα metres.

Height of the cliff=ABIn triangle EFC,EC=k=> EF=ksinϕ,CF=kcosϕ=BDIn triangle ABC, Let AB=x=>AC=xcosecθ;BC=xcotθFB=BC−CF=xcotθ−kcosϕ=EDAD=EDtanα=tanα(xcotθ−kcosϕ)AD=AB−BD=x−ksinϕ=>tanα=x−ksinαxcotθ−kcosϕ