Question

A vertical tower stands on a horizontal plane and is surmounted by a vertical flagstaff of height h. At a point on the plane, the angles of elevation of the top and bottom of the flagstaff are $\theta$ and $\varphi$ respectively. Find the height of the tower.

Answer 1

Let distance between the foot of tower and point of observation

$CD=x$

$y=$ height of tower

In $\Delta ACD$

$\cot \theta =\frac{x}{h+y}$

$\left(h+y\right)\cot \theta =x$...............(1)

IN $\Delta BCD$

$\cot \varphi =\frac{x}{y}$

$x=y\cot \varphi$......................... (2)

From (1) & (2) $\left(h+y\right)\cot \theta =y\cot \varphi$

$h\cot \theta =y(\cot \theta -\cot \varphi )$

$y=\frac{h\cot \theta }{\cot \varphi -\cot \theta }=\frac{h}{\tan \theta }\left(\frac{\tan \theta \tan \varphi }{\tan \theta -\tan \varphi }\right)$

(height of tower ) $y=\frac{h\cos \varphi }{\tan \theta -\tan \varphi }$