Question

The angle of elevation of a cliff from a fixed point `A` is `theta`. After going up a distance of `'k' m` towards the top of the cliff, at an angle of `phi`, it is found that the angle of elevation is `alpha`. Find height of cliff in terms of `k`?

Answer 1

Let CB be the cliff . From point A the angle of elevation of the peak C of the cliff is `/_CAB=theta`'. After going up a distance , `AD=k` `m` towards the top of the cliff at an angle,`/_DAE =phi`,it ls found the angle of elevation `/_CDF=alpha`.

DF and DE are perpendiculars drawn from D on CB and AB.

Now `DE =ksinphi and AE =kcosphi`

Let
`h="CB the height of the cliff" and b=BA`

For `DeltaCAB," "(CB)/(BA)=tantheta`

`=>b/h=cottheta=>b=hcottheta`

Now `DF=BE=BA-AE=b-kcosphi`

`CF=CB-FB=CB-DE=h-ksinphi`

For `DeltaCDF," "(CF)/(DF)=tanalpha`

`=>(h-ksinphi)/(b-kcosphi)=tanalpha`

`=>(h-ksinphi)/(hcottheta-kcosphi)=tanalpha`

`=>h-htanalphacottheta=ksinphi-kcosphitanalpha`

`=>h=(k(sinphi-cosphitanalpha))/(1-tanalphacottheta)`

`=>h=(ksintheta(sinphicosalpha-cosphisinalpha))/(sinthetacosalpha-costhetasinalpha)`

`=>h=(ksinthetasin(phi-alpha))/(sin(theta-alpha))`

`=>h=(ksinthetasin(alpha-phi))/(sin(alpha-theta))`