Question

Hari, standing on the top of a building, sees the top of a tower at an angle of elevation of ${50}^{\circ }$ and the foot of the tower at an angle of depression of ${20}^{\circ }$. Height of Hari is $1.6$ metre and height of the building on which he is standing is $9.2$ metre.
(a) Draw a rough sketch according to the given information.
(b) How far is the tower from the building?
(c) Calculate the height of the tower.
$\sin {20}^{\circ }=0.34,\cos {20}^{\circ }=0.94,\tan {20}^{\circ }=0.36$,
$\sin {50}^{\circ }=0.77,\cos {50}^{\circ }=0.64,\tan {50}^{\circ }=1.19$.

Answer 1

(a)

The rough sketch is as follows:
Hari is standing at $D$ and his height is $1.6\text{m}$.
Height of the building is $DF=9.2\text{m}$.
The angle of elevation of the top of the tower is ${50}^{\circ }$ and the angle of depression of the foot of the tower is ${20}^{\circ }$.

(b)
The distance between the tower and the building is $BC=DE=FG$
Consider $△BCG$:
$\tan {20}^{\circ }=\frac{CG}{BC}\Rightarrow BC=\frac{CG}{\tan {20}^{\circ }}$
$\Rightarrow BC=\frac{CE+EG}{0.36}=\frac{1.6+9.2}{0.36}=30\text{m}$

(c) The height of the tower is $AG=AC+CE+EG=AC+1.6+9.2=AC+10.8$
Consider $△ABC$, let $AC=h$
$\tan {50}^{\circ }=\frac{AC}{BC}$
$\Rightarrow 1.19=\frac{h}{30}$
$\Rightarrow h=35.7\text{m}$
Thus, total height of the tower $=35.7+10.8=46.5\text{m}$