Question

A vertical tower stands on the horizontal ground and is surmounted by a vertical flagstaff of height 'h' metre. At a point on the ground, the angle of elevation of the bottom of the flagstaff is $\alpha$ and that of the top of the flagstaff is $\beta$. Find the height of the tower.

Answer 1

Let the height of the tower be 'x' and the distance of the point from the bottom of the tower be 'y'.

In $\Delta ABD$

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

$y=\frac{x}{tan\alpha }$ ....... (i)

In $\Delta ACD$

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

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

$tan\beta =\frac{(x+h)tan\alpha }{x}$

$\therefore xtan\beta =xtan\alpha +htan\alpha$

$x(tan\beta -tan\alpha )=htan\alpha$

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