A statue, $1.6\ m$ tall, stands on the top of a pedestal. From a point on the ground, the angle of elevation of the top of the statue is $60^o$ and from the same point the angle of elevation of the top of the pedestal is $45^o$. Find the height of the pedestal.
Given:
A statue \( 1.6 \mathrm{~m} \) tall stands on the top of pedestal.
From a point on the ground, the angle of elevation of the top of the statue is \( 60^{\circ} \) and from the same point the angle of elevation of the top of the pedestal is \( 45^{\circ} \).
To do:
We have to find the height of the pedestal.
Solution:
Let $AB$ be the height of the pedestal and $BC$ be the height of the statue.
Point $D$ be the point of observation.
From the figure,
$\mathrm{BC}=1.6 \mathrm{~m}, \angle \mathrm{CDA}=60^{\circ}, \angle \mathrm{BDA}=45^{\circ}$
Let the height of the pedestal be $\mathrm{AB}=h \mathrm{~m}$ and the distance of the pedestal from the point $D$ be $\mathrm{DA}=x \mathrm{~m}$.
This implies,
$\mathrm{AC}=1.6+h \mathrm{~m}$
We know that,
$\tan \theta=\frac{\text { Opposite }}{\text { Adjacent }}$
$=\frac{\text { BA }}{DA}$
$\Rightarrow \tan 45^{\circ}=\frac{h}{x}$
$\Rightarrow 1=\frac{h}{x}$
$\Rightarrow x(1)=h \mathrm{~m}$
$\Rightarrow x=h \mathrm{~m}$.........(i)
Similarly,
$\tan \theta=\frac{\text { Opposite }}{\text { Adjacent }}$
$=\frac{\text { AC }}{DA}$
$\Rightarrow \tan 60^{\circ}=\frac{1.6+h}{x}$
$\Rightarrow \sqrt3=\frac{1.6+x}{x}$ [From (i)]
$\Rightarrow x\sqrt3=1.6+x \mathrm{~m}$
$\Rightarrow x(\sqrt3-1)=1.6 \mathrm{~m}$
$\Rightarrow x=\frac{1.6}{\sqrt3-1)} \mathrm{~m}$
$\Rightarrow x=\frac{1.6\times(\sqrt3+1)}{(\sqrt3-1)(\sqrt3+1)} \mathrm{~m}$
$\Rightarrow x=\frac{1.6(\sqrt3+1)}{3-1} \mathrm{~m}$
$\Rightarrow x=\frac{1.6(\sqrt3+1)}{2} \mathrm{~m}$
$\Rightarrow x=\frac{4(\sqrt3+1)}{5} \mathrm{~m}$
Therefore, the height of the pedestal is $\frac{4(\sqrt3+1)}{5} \mathrm{~m}$.
Related Articles
- A statue \( 1.6 \mathrm{~m} \) tall stands on the top of pedestal. From a point on the ground, the angle of elevation of the top of the statue is \( 60^{\circ} \) and from the same point the angle of elevation of the top of the pedestal is \( 45^{\circ} \). Find the height of the pedestal.
- 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 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 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. $( Use\ \sqrt{3} \ =\ 1.73)$
- 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.
- 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.
- From a point on the ground, the angles of elevation of the bottom and the top of a transmission tower fixed at the top of a $20\ m$ high building are $45^o$ and $60^o$ respectively. 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.
- 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.
- 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 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 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.
- 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."
- 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.
- 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.
Kickstart Your Career
Get certified by completing the course
Get Started