Question

A vertical tower stands on a horizontal plane and is surmounted by a flagstaff of height $7$ m. From a point on the plane, the angle of elevation of the bottom of the flag-staff is ${30}^{\circ }$ and that of the top of the flag-staff is ${45}^{\circ }$. Find the height of the tower.

Answer 1

$\Rightarrow$ In $△CBD,\tan {30}^o=\frac{BD}{BC}$

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

$\Rightarrow$ $x=\sqrt{3}h$ ----- ( 1 )

$\Rightarrow$ In$△ABC,\tan {45}^o=\frac{AB}{BC}$

$\Rightarrow$ $1=\frac{7+h}{x}$

$\Rightarrow$ $x=7+h$

$\Rightarrow$ $\sqrt{3}h=7+h$ [ From ( 1 ) ]

$\Rightarrow$ $\sqrt{3}h-h=7$

$\Rightarrow$ $(\sqrt{3}-1)h=7$

$\Rightarrow$ $h=\frac{7}{\sqrt{3}-1}$

$\Rightarrow$ $h=\frac{7}{\sqrt{3}-1}\times \frac{\sqrt{3}+1}{\sqrt{3}+1}$

$\therefore$ $h=\frac{7(\sqrt{3}+1)}{3-1}=\frac{7(1.73+1)}{2}=9.55m$

$\therefore$ Height of the tower is $9.55m$.