The angle of elevation of the top of the building from the foot of the tower is $ 30^{\circ} $ and the angle of the top of the tower from the foot of the building is $ 60^{\circ} $. If the tower is $ 50 \mathrm{~m} $ high, find the height of the building.

Given:

The angle of elevation of the top of the building from the foot of the tower is \( 30^{\circ} \) and the angle of the top of the tower from the foot of the building is \( 60^{\circ} \).

The tower is \( 50 \mathrm{~m} \) high.

To do:

We have to find the height of the building.

Solution:

Let $AB$ be the height of the tower and $CD$ be the height of the building.

From the figure,

$\mathrm{AB}=50 \mathrm{~m}, \angle \mathrm{BCA}=60^{\circ}, \angle \mathrm{DAC}=30^{\circ}$

Let the height of the building be $\mathrm{CD}=h \mathrm{~m}$ and the distance between the building and the tower be $\mathrm{CA}=x \mathrm{~m}$.

We know that,

$\tan \theta=\frac{\text { Opposite }}{\text { Adjacent }}$

$=\frac{\text { AB }}{CA}$

$\Rightarrow \tan 60^{\circ}=\frac{50}{x}$

$\Rightarrow \sqrt3=\frac{50}{x}$

$\Rightarrow x=\frac{50}{\sqrt3} \mathrm{~m}$.........(i)

Similarly,

$\tan \theta=\frac{\text { Opposite }}{\text { Adjacent }}$

$=\frac{\text { DC }}{AC}$

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

$\Rightarrow \frac{1}{\sqrt3}=\frac{h}{\frac{50}{\sqrt3}}$ [From (i)]

$\Rightarrow \frac{1}{\sqrt3}\times\frac{50}{\sqrt3}=h \mathrm{~m}$

$\Rightarrow h=\frac{50}{3} \mathrm{~m}$

Therefore, the height of the building is $\frac{50}{3} \mathrm{~m}$.

Related Articles The angle of elevation of the top of a building from the foot of a tower is $30^o$ and the angle of elevation of the top of the tower from the foot of the building is $60^o$. If the tower is $50\ m$ high, find the height of the building.
From the top of a building \( 15 \mathrm{~m} \) high the angle of elevation of the top of a tower is found to be \( 30^{\circ} \). From the bottom of the same building, the angle of elevation of the top of the tower is found to be \( 60^{\circ} \). Find the height of the tower and the distance between the tower and building.
From the top of a \( 7 \mathrm{~m} \) high building, the angle of elevation of the top of a cable tower is \( 60^{\circ} \) and the angle of depression of its foot is \( 45^{\circ} . \) Determine the height of the tower.
The angle of elevation of the top of a hill at the foot of a tower is \( 60^{\circ} \) and the angle of elevation of the top of the tower from the foot of the hill is \( 30^{\circ} \). If the tower is \( 50 \mathrm{~m} \) high, what is the height of the hill?
From the top of a 7 m high building, the angle of the elevation of the top of a tower is $60^{o}$ and the angle of the depression of the foot of the tower is $30^{o}$. Find the height of the tower.
From the top of a $7\ m$ high building, the angle of elevation of the top of a cable tower is $60^o$ and the angle of depression of its foot is $45^o$. Determine the height of the tower.
A person observed the angle of elevation of the top of a tower as \( 30^{\circ} \). He walked \( 50 \mathrm{~m} \) towards the foot of the tower along level ground and found the angle of elevation of the top of the tower as \( 60^{\circ} \). Find the height of the tower.
A tower stands vertically on the ground. From a point on the ground, \( 20 \mathrm{~m} \) away from the foot of the tower, the angle of elevation of the top of the tower is \( 60^{\circ} \). What is the height of the tower?
The angle of elevation of the top of a tower $30\ m$ high from the foot of another tower in the same plane is $60^o$ and the angle of elevation of the top of the second tower from the foot of the first tower is $30^o$. then find the distance between the two towers.
The angle of elevation of the top of a tower from a point on the ground, which is $30\ m$ away from the foot of the tower is $30^o$. Find the height of the tower.
The angle of elevation of a tower from a point on the same level as the foot of the tower is \( 30^{\circ} \). On advancing 150 metres towards the foot of the tower, the angle of elevation of the tower becomes \( 60^{\circ} \). Show that the height of the tower is \( 129.9 \) metres (Use \( \sqrt{3}=1.732 \) ).
From a point on the ground the angles of elevation of the bottom and top of a transmission tower fixed at the top of \( 20 \mathrm{~m} \) high building are \( 45^{\circ} \) and \( 60^{\circ} \) respectively. Find the height of the transimission tower.
The angle of elevation of the top of tower, from the point on the ground and at a distance of 30 m from its foot, is 30o. Find the height of tower.
The angle of elevation of the top of a tower from a point \( A \) on the ground is \( 30^{\circ} \). On moving a distance of 20 metres towards the foot of the tower to a point \( B \) the angle of elevation increases to \( 60^{\circ} \). Find the height of the tower and the distance of the tower from the point \( A \).
The angles of elevation of the top of a rock from the top and foot of a \( 100 \mathrm{~m} \) high tower are respectively \( 30^{\circ} \) and \( 45^{\circ} \). Find the height of the rock.
Kickstart Your Career
Get certified by completing the course

Get Started