Question

A tower subtends an angle $\theta$ at a point A in the plane of its base and the angle of depression of the foot of the tower at a point b m just above A is $\varphi$ . Find the height of the tower.

Answer 1

To find height of tower=CD

In $\Delta BCE$,

$\tan \varphi =\frac{EC}{BE}.........\left(\#\right)$

Now, ABEC is a rectangle. Thus:

i) EC=BA

ii) BE=AC

$\tan \varphi =\frac{b}{AC}$, from above (#)

$\Rightarrow AC=\frac{b}{\tan \varphi }.....(\ast )$

In $\Delta ACD$ ,

$\tan \theta =\frac{\text{CD}}{\text{AC}}$

$\Rightarrow \text{CD}=\frac{\text{b}}{tan\varphi }tan\theta \text{,}\text{from}\text{above}(\ast )$

$heightoftower=b\frac{\tan \theta }{\tan \varphi }$