A TV tower stands vertically on a bank of a canal. From a point on the other bank directly opposite the tower, the angle of elevation of the top of the tower is $60^o$. From another point $20\ m$ away from this point on the line joining this point to the foot of the tower, the angle of elevation of the top of the tower is $30^o$ (see the given figure). Find the height of the tower and the width of the canal.
"
Given:
A TV tower stands vertically on a bank of a canal. From a point on the other bank directly opposite the tower, the angle of elevation of the top of the tower is $60^o$. From another point $20\ m$ away from this point on the line joining this point to the foot of the tower, the angle of elevation of the top of the tower is $30^o$.
To do:
We have to find the height of the tower and the width of the canal.
Solution:
Let $AB$ be the height of the tower and $BC$ be the width of the canal.
Let point $C$ be the point of observation on the other bank and point $D$ be \( 20 \mathrm{~m} \) away from point $C$ on the same bank.
From the figure,
$\mathrm{CD}=20 \mathrm{~m}, \angle \mathrm{ADB}=30^{\circ}, \angle \mathrm{ACB}=60^{\circ}$
Let the height of the tower be $\mathrm{AB}=h \mathrm{~m}$ and the width of the canal be $\mathrm{BC}=x \mathrm{~m}$.
This implies,
$\mathrm{DB}=20+x \mathrm{~m}$
We know that,
$\tan \theta=\frac{\text { Opposite }}{\text { Adjacent }}$
$=\frac{\text { AB }}{BC}$
$\Rightarrow \tan 60^{\circ}=\frac{h}{x}$
$\Rightarrow \sqrt3=\frac{h}{x}$
$\Rightarrow x(\sqrt3)=h \mathrm{~m}$
$\Rightarrow h=x\sqrt3 \mathrm{~m}$.........(i)
Similarly,
$\tan \theta=\frac{\text { Opposite }}{\text { Adjacent }}$
$=\frac{\text { AB }}{DB}$
$\Rightarrow \tan 30^{\circ}=\frac{h}{x+20}$
$\Rightarrow \frac{1}{\sqrt3}=\frac{x\sqrt3}{x+20}$ [From (i)]
$\Rightarrow x+20=x\sqrt3(\sqrt3) \mathrm{~m}$
$\Rightarrow 3x-x=20 \mathrm{~m}$
$\Rightarrow x=\frac{20}{2} \mathrm{~m}$
$\Rightarrow x=10 \mathrm{~m}$
$\Rightarrow h=10\sqrt3 \mathrm{~m}$
Therefore, the height of the tower is $10\sqrt3 \mathrm{~m}$ and the width of the canal is $10 \mathrm{~m}$.
Related Articles
- A T.V. Tower stands vertically on a bank of a river. From a point on the other bank directly opposite the tower, the angle of elevation of the top of the tower is \( 60^{\circ} \). From a point \( 20 \mathrm{~m} \) away this point on the same bank, the angle of elevation of the top of the tower is \( 30^{\circ} \). Find the height of the tower and the width of the river.
- 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 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 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 angle of the elevation of the top of vertical tower from a point on the ground is 60°. From another point 10 m vertically above the first, its angle of elevation is 30°. Find the height of the tower.
- The angle of elevation of top of tower from certain point is $30^o$. if the observer moves $20\ m$ towards the tower, the angle of elevation of the top increases by $15^o$. Find the height of the 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.
- 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.
- 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.
- 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 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 \( \sqrt{3}=1.732 \) ).
- The angle of elevation of the top Q of a vertical tower PQ from a point X on the ground is $60^{o}$. From a point Y, $40\ m$ vertically above X, the angle of elevation of the top Q of tower is $45^{o}$. Find the height of the tower PQ and the distance PX. $( Use\ \sqrt{3} \ =\ 1.73)$
- A flag-staff stands on the top of 5 m high tower. From a point on the ground, the angle of elevation of the top of the flag-staff is \( 60^{\circ} \) and from the same point, the angle of elevation of the top of the tower is \( 45^{\circ} \). Find the height of the flag-staff.
- The angle of elevation of the top of a vertical tower \( P Q \) from a point \( X \) on the ground is \( 60^{\circ} \). At a point \( Y, 40 \) m vertically above \( X \), the angle of elevation of the top is \( 45^{\circ} \). Calculate the height of the tower.
- A tower stands vertically on the ground. From a point on the ground which is 25 in away in the foot of the tower, the angle of the elevation of the top of the tower is found to be $45^{o}$. Then the height $(in\ meter)$ of the tower is:$( A) \ 25\sqrt{2}$$( B) \ 25\sqrt{3}$$( C) \ 25$$( D) \ 12.5$
Kickstart Your Career
Get certified by completing the course
Get Started