Angle of Depression

Q.

From the top of a building AB, 60 m high, the angles of depression of the top and bottom of a vertical lamp-post CD are observed to be ${30}^{\circ }$ and ${60}^{\circ }$ respectively. Find

(i) the horizontal distance between AB and CD,

(ii) the height of the lamp-post,

(iii) the difference between the heights of the building and the lamp-post.

Q.

A triangle ABC is right-angled at B. AB = 6 cm and BC = 8 cm. What is the length of AC?

1. 2 cm

2. 10 cm

3. 4 cm

4. 5 cm

Q.

A man standing on the deck of a ship, which is 10 m above the water level, observes the angle of elevation of the top of a hill as ${60}^{\circ }$, and the angle of depression of the base of the hill as ${30}^{\circ }$. Find the distance of the hill from the ship and the height of the hill.

Q. The lower window of a house is at a height of 2 m above the ground and its upper window is 4 m vertically above the lower window . At certain instant the angles of elevation of a balloon from these windows are observed to be 60⁰ and 30⁰ respectively. Find the height of the balloon above the ground.

Q.

The angles of depression of two ships from the top of a lighthouse and on the same side of it are found to be ${45}^{\circ }$ and ${30}^{\circ }$ respectively. If the ships are $200m$ apart, find the height of the lighthouse.

Q.

The angles of depression of the top and bottom of a tower as seen from the top of a $60\sqrt{3}$-m-high cliff are ${45}^{\circ }and{60}^{\circ }$ respectively. Find the height of the tower.

Q. As observed from the top of a 100 m high light house from the sea-level, the angles of depression of two ships are ${30}^o$ and ${45}^o$. If one ship is exactly behind the other on the same side of the light house, find the distance between the two ships. [Use $\sqrt{3}=1.732$]

Q.

A tower subtends an angle of ${30}^{\circ }$ at a point, C, on the same level as its foot, A. At a second point, D, 'h' meters above C, the angle of depression of the foot of the tower is ${60}^{\circ }$. What is the height of the tower?

1. $\frac{h}{2}m$

2. $\frac{h}{3}m$

3. $h\sqrt{3}m$

4. $\frac{h}{\sqrt{3}}m$

Q. In a triangle ABC, right angled at B, if AB = 3 cm, BC = 4 cm, then the value of AC (in cm) is .

1. 2
2. 3
3. 4
4. 5

Q. The angle of elevation of the top of a tower as observed form a point in a horizontal plane through the foot of the tower is 32°. When the observer moves towards the tower a distance of 100 m, he finds the angle of elevation of the top to be 63°. Find the height of the tower and the distance of the first position from the tower. [Take tan 32° = 0.6248 and tan 63° = 1.9626]

Q.

The angle of elevation of the top of a chimney from the top of a tower is ${60}^{\circ }$ and the angle of depression of the foot of the chimney from the top of the tower is ${30}^{\circ }$. If the height of the tower is $40m$, find the height of the chimney. According to pollution control norms, the minimum height of a smoke emitting chimney should be $100m$. State if the height of the above mentioned chimney meets the polution norms. What valued is discussed in this question?

Q.

From the top of a lighthouse, the angles of depression of two ships on the opposite sides of it are observed to be $\alpha$ and $\beta$. If the height of the lighthouse be h metres and the line joining the ships passes through the foot of the lighthouse, show that the distance between the ships is $\frac{h(tan\alpha +tan\beta )}{tan\alpha tan\beta }$ metres. [3 MARKS]

Q. If the angle of elevation of a cloud from a point 200 mabove a lake is ${30}^{\circ }$and the angle of depression of the reflection of the cloud in the lake from the same point is ${60}^{\circ }$, then the height of the cloud above the lake is:

1. 500 m
2. 30 m
3. 400 m
4. 200 m

Q. Question 12
To warn ships for underwater rocks, a lighthouse spreads a red coloured light over a sector of angle ${80}^{\circ }$ to a distance of 16.5 km. Find the area of the sea over which the ships are warned.
$(Use\pi =3.14)$

Q.

A 1.2m tall girl spots a balloon moving with the wind in a horizontal line at a height of 88.2m from the ground. The angle of elevation of the balloon from the eyes of the girl at any instant is ${60}^{\circ }$. After sometime, the angle of elevation reduces to ${30}^{\circ }$. Find the distance travelled by the balloon during the interval?

1. 48√3m

2. 58√3m

3. 52√3m

4. 65√2m

Q. It is found that on walking x meters towards a chimney in a horizontal line through its base, the elevation of its top changes from 30° to 60°. The height of the chimney is

(a) $3\sqrt{2}x$

(b) $2\sqrt{3}x$

(c) $\frac{\sqrt{3}}{2}x$

(d) $\frac{2}{\sqrt{3}}x$

Q. The angle of elevation of a tower from a point on the same level as the foot of the tower is 30°. On advancing 150 metres towards the foot of the tower, the angle of elevation of the tower becomes 60°. Show that the height of the tower is 129.9 metres (Use $\sqrt{3}=1.732$)

Q. The angles of elevation of the top of a tower from two points at distances of 5 metres and 20 metres from the base of the tower and in the same straight line with it, are complementary. Find the height of the tower. [CBSE 2014]

Q. The angle of elevation of the top of a tower at a distance of 120 m from a point A on the ground is 45°. If the angle of elevation of the top of a flagstaff fixed at the top of the tower, at A is 60°, then find the height of the flagstaff. [Use $\sqrt{3}$ = 1.732] [CBSE 2014]

Q. A person observed the angle of elevation of the top of a tower as 30°. He walked 50 m towards the foot of the tower along level ground and found the angle of elevation of the top of the tower as 60°. Find the height of the tower.

Q. An observer , 1.7 m tall , is 20$\sqrt{3}$ m away from a tower . The angle of elevation from the eye of an observer to the top of tower is 30⁰ . Find the height of the tower.

Q.

The horizontal distance between two trees of different heights is 60 m. The angle of depression of the top of the first tree whenseen from the top of the second tree is ${45}^{\circ }$. If the height of the second tree is 80 m, find the height of the first tree.

Q.

From a lighthouse, the angles of depression of two ships on opposite sides of the lighthouse were observed to be $30°$ and $45°$ If the lighthouse is $90m$ high and the line joining the two ships passes through the foot of the lighthouse, find the distance between the two ships.

Q. Question 14
From the top of a tower h m high angles of depression of two objects, which are in line with foot of the tower are $\alpha$ and $\beta (\beta >\alpha )$. Find the distance between the two objects.

Q.

A man on the top of a tower, standing on the sea shore, finds a boat coming towards him takes 10 minutes for the angle of depression to change from ${30}^{\circ }$ to ${60}^{\circ }$. How soon will the boat reach the sea-shore?

1. 20 min

2. 10 min

3. 5 min

4. 2.5 min

Q. The angles of depression of the top and bottom of a tower as seen from the top of a $60\sqrt{3}$ m high cliff are 45° and 60° respectively. Find the height of the tower.

Q.

From the top of a 120m high tower, a man observes two cars on the opposite sides of the tower and in straight line with the base of tower with angles of depression as ${60}^{\circ }$ and ${45}^{\circ }$. Find the distance between the cars. (Take $\sqrt{3}=1.732)$

Q.

The combined equation of the bisectors of the angle between the lines represented by $\sqrt{3}(x^2+y^2)=4xy$.

1. $y^2-x^2=0$

2. $xy=0$

3. $x^2+y^2=2xy$

4. None of these

Q.

From the top of a 50 m high tower, the angles of depression of the top and bottom of a pole are observed to be ${45}^{\circ }$ and ${60}^{\circ }$ respectively. Find the height of the pole.

Q. From a point on the ground 40 m away from the foot of a tower, the angle of elevation of the top of the tower is 30°. The angle of elevation of the top of a water tank (on the top of the tower) is 45°. Find (i) the height of the tower, (ii) the depth of the tank.