The angle of elevation of the top of a tower from a point on the ground 30 m away from the foot of the tower is 30°.
Question:
The angle of elevation of the top of a tower from a point on the ground 30 m away from the foot of the tower is 30°. The height of the tower is
(a) 30 m
(b) $10 \sqrt{3} \mathrm{~m}$
(c) 20 m
(d) $10 \sqrt{2} \mathrm{~m}$
Solution:
(b) $10 \sqrt{3} \mathrm{~m}$
Let $A B$ be the tower and $O$ be the point of observation.
Also,
$\angle A O B=30^{\circ}$ and $O B=30 \mathrm{~m}$
Let:
$A B=h \mathrm{~m}$
In $\triangle A O B$, we have:
$\frac{A B}{O B}=\tan 30^{\circ}=\frac{1}{\sqrt{3}}$
$\Rightarrow \frac{h}{30}=\frac{1}{\sqrt{3}}$
$\Rightarrow h=\frac{30}{\sqrt{3}} \times \frac{\sqrt{3}}{\sqrt{3}}=\frac{30 \sqrt{3}}{3}=10 \sqrt{3} \mathrm{~m}$
Hence, the height of the tower is $10 \sqrt{3} \mathrm{~m}$.