Question

A vertical tower stands on a horizontal plane and is surmounted by a vertical flag staff of height $5$ meters. At a point on the plane the angle of elevation of the bottom and the top of the flag staff are respectively ${30}^{\circ }$ and ${60}^{\circ }$. Find the height of tower.

Answer 1

Let the height of the tower be ${}^{\prime }{h}^{\prime }\text{m}.$

Now, In $△BCA$

$\tan 60°=\frac{AC}{BC}=\frac{(5+h)}{BC}$

$\sqrt{3}=\frac{(5+h)}{BC}$

$\Rightarrow BC=\frac{(5+h)}{\sqrt{3}}⟶\left(1\right)$

Now, In $△DBC,$

$\tan 30°=\frac{CD}{BC}=\frac{h}{BC}$

$\frac{1}{\sqrt{3}}=\frac{h}{BC}$

$\Rightarrow BC=h\sqrt{3}⟶\left(2\right)$

From equation $\left(1\right)\&\left(2\right),$ we get

$\frac{(5+h)}{\sqrt{3}}=h\sqrt{3}$

$5+h=3h$

$2h=5$

$\Rightarrow h=2.5\text{m}$

$\therefore$ Height of the tower is $2.5\text{m}.$

Hence, the answer is $2.5\text{m}.$