A vertical tower stands on a horizontal plane and is surmounted by a vertical flag-staff

Question:

A vertical tower stands on a horizontal plane and is surmounted by a vertical flag-staff of height 5 metres. At a point on the plane, the angles of elevation of the bottom and the top of the flag-staff are respectively 30° and 60°. Find the height of the tower.

Solution:

Let BC be the tower of height m and AB be the flag staff with distance 5m.Then angle of elevation from the top and bottom of flag staff are 60° and 30° respectively.

Let $C D=x$ and $\angle A D C=60^{\circ}, \angle B D C=30^{\circ}$

Here we have to find height of tower.

So we use trigonometric ratios.

In a triangle,

$\Rightarrow \quad \tan D=\frac{B C}{C D}$

$\Rightarrow \quad \tan 30^{\circ}=\frac{h}{x}$

$\Rightarrow \quad \frac{1}{\sqrt{3}}=\frac{h}{x}$

$\Rightarrow \quad x=\sqrt{3} h$

Again in a triangle $A C D$

$\Rightarrow \quad \tan D=\frac{A B+B C}{C D}$

$\Rightarrow \quad \tan 60^{\circ}=\frac{h+5}{x}$

$\Rightarrow \quad \sqrt{3}=\frac{h+5}{x}$

$\Rightarrow \quad \sqrt{3} x=h+5$

$\Rightarrow \sqrt{3} \times h \sqrt{3}=h+5$

$\Rightarrow \quad 3 h=h+5$

$\Rightarrow \quad 2 h=5$

$\Rightarrow \quad h=2.5$

Hence the height of tree is $2.5 \mathrm{~m}$.