The angles of elevation of the top of a cliff from two points on the ground, 200 m apart, and on opposite sides of the cliff, are \(30^\circ\) and \(45^\circ\). What is the height of the cliff?

Let the height of the cliff = h m
Let distances of the two points from the foot of the cliff be $ x $ and y
x + y = 200
$\tan 30^\circ = \frac{h}{x} = \frac{1}{\sqrt{3}}=\frac{h}{x} = x=\sqrt{3}h$
$\tan 45^\circ = \frac{h}{y} = 1=\frac{h}{y} = y=h$
$x+y=200 = \sqrt{3}h + h = 200$
$h(\sqrt{3}+1)=200$
$h=\frac{200}{\sqrt{3}+1}$
rationalising
$h=\frac{200(\sqrt{3}-1)}{2}$
$h=100(\sqrt{3}-1)\ \text{m}$