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. $( Use\ \sqrt{3} \ =\ 1.73)$
Given: 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 $45^{o}$.
To do: To find the height of the tower PQ and the distance PX.
Solution:
$MP= YX=40\ m$
$\therefore QM = h –40$
In right angled $\vartriangle QMY$,
$tan45^{o} =\frac{QM}{MY} =1=\frac{h-40}{PX} \ \ \ \ \ \ \ \ ( \because \ MY\ =\ PX)$
$\therefore PX = h –40 \ \ \ \ \ \ ....( 1)$
In right angled $\vartriangle QPX$,
$tan\ 60^{o} =\frac{OP}{PX} =\sqrt{3} =\frac{h}{PX} =\frac{h}{h-40} \ \ \ ....( 2)$
$\Rightarrow \frac{h}{h-40} =\sqrt{3}$
$\Rightarrow h=h\sqrt{3} -40\sqrt{3}$
$\Rightarrow h\sqrt{3} -h=40\sqrt{3}$
$\Rightarrow h\times 1.73-h=40\times 1.73$
$\Rightarrow 0.73h=69.2$
Or $h=\frac{69.2}{0.73}$
Thus, $PQ$ is $94.79\ m$.
Related Articles
- 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 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 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.
- 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 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.
- 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 \( A \).
- 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."
- 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.
- 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.
- 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.
- 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.
- 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 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.
Kickstart Your Career
Get certified by completing the course
Get Started