Question

A vertical tower stands on horizontal plane and is surmounted by a vertical flagstaff of height h metre. At a point on the plane, the angle of elevation of the bottom of the flagstaff is $\alpha$ and that of the top of flagstaff is $\beta$. Prove that the height of the tower is
$\frac{h\tan \alpha }{\tan \beta -\tan \alpha }$

Answer 1

$\tan \alpha =\frac{x}{z}$

$\Rightarrow z=x\cot \alpha$

$\tan \beta =\frac{x+h}{z}$

$\Rightarrow z=(x+h)\cot \beta$

Hence $(x+h)\cot \beta =x\cot \alpha$

Solving this we get

$x=\frac{h\tan \alpha }{\tan \beta -\tan \alpha }$