## Chapter 14 – Height and Distance Exercise Ex. 14

A tower stands vertically on the ground. From a point on the ground which is 20 m away from the foot of the tower, the angle of elevation of its top is found to be 60°. Find the height of the tower. [Take ]

Let AB be the tower standing on a level ground and O be the position of the observer. Then OA = 20 m and _{}OAB = 90° and _{}AOB = 60°

Let AB = h meters

From the right OAB, we have

_{}

Hence the height of the tower is _{}

A kite is flying at a height of 75m from the level ground, attached to a string inclined at 60° to the horizontal. Find the length of the string assuming that there is no slack in it. _{}

Let OB be the length of the string from the level of ground and O be the point of the observer, then, AB = 75m and _{}OAB = 90° and _{}AOB = 60°, let OB = l meters.

From the right OAB, we have

_{}

An observer 1.5 m tall is 30 m away from a

chimney. The angle of elevation of the top of the chimney from his eye is

60°. Find the height of the chimney.

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.

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.

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.

A vertical tower stands on a horizontal

plane and is surmounted by a vertical flagstaff of height 6 m. At a point on

the plane, the angle of elevation of the bottom of the flagstaff is 30° and that of the top of the flagstaff is 60°. Find the height of the tower.

A statue 1.46 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° and from the same point, the angle of elevation of the top of the pedestal is 45°. Find the height of the pedestal. _{}

Let SP be the statue and PB be the pedestal. Angles of elevation of S and P are 60° and 45° respectively.

Further suppose AB = x m, PB = h m

In right _{}ABS,

_{}

In right _{}PAB,

_{}

Thus, height of the pedestal = 2m

The angle of elevation of the top of an unfinished tower at a distance of 75m from its base is 30°. How much higher must the tower be raised so that the angle of elevation of its top at the same point may be 60°?_{}

Let AB be the unfinished tower and let AC be complete tower.

Let O be the point of observation. Then,

OA = 75 m

_{}AOB = 30° and _{}AOC = 60°

Let AB = h meters

And AC = H meters

_{}

Hence, the required height is

On a horizontal plane there is a vertical tower with a flagpole on the top of the tower. At a point, 9 metres away from the foot of the tower, the angle of elevation of the top and bottom of the flagpole are 60° and 30° respectively. Find the height of the tower and the flagpole mounted on it. _{}

Let AB be the tower and BC be flagpole, Let O be the point of observation.

Then, OA = 9 m, _{}AOB = 30° and _{}AOC = 60°

From right angled BOA

_{}

From right angled OAC

_{}

Thus _{}

Hence, height of the tower= 5.196 m and the height of the flagpole = 10.392 m

Two poles of equal heights are standing opposite to each other on either side of the road which is 80 m wide. From a point P between them on the road, the angle of elevation of the top of one pole is 60° and the angle of depression from the top of another pole at P is 30°. Find the height of each pole and distances of the point P from the poles.

Two men are on opposite sides of a tower.

They measure the angles of elevation of the top of the tower as 30° and 45°

respectively. If the height of the tower is 50 metres,

find the distance between the two men.

From the top of a tower 100 m high, a man

observes two cars on the opposite sides of the tower with angles of

depression 30° and 45° respectively. Find the distance between the cars.

A straight highway leads to the foot of a

tower. A man standing on the top of the tower observes a car at an angle of

depression of 30°, which is approaching the foot of the tower with a uniform

speed. Six seconds later, the angle of depression of the car is found to be

60°. Find the time taken by the car to reach the foot of the tower form this

point.

A TV tower stands vertically on a bank of

canal. From a point on the other bank directly opposite the tower, the angle

of elevation of the top of the tower is 60°. 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°. Find the height of the

tower and the width of the canal.

The angle of elevation of the top of a

building from the foot of a tower is 30°. The angle of elevation of the top

of the tower from the foot of the building is 60°. If the tower is 60 m high,

find the height of the building.

The horizontal distance between two towers is 60metres. The angle of depression of the top of the first tower when seen from the top of the second tower is 30°. If the height of the second tower is 90metres, find the height of the tower.

Let AB and CD be the first and second towers respectively.

Then, CD = 90 m and AC = 60 m.

Let DE be the horizontal line through D.

Draw BF _{}CD,

Then, BF = AC = 60 m

_{}FBD = _{}EDB = 30°

_{}

The angle of elevation of the top of a

chimney from the foot of a tower is 60° and the angle of depression of the

foot of the chimney from the top of the tower is 30°. If the height of the

tower is 40 metres, find the height of the chimney.

According to pollution control norms, the

minimum height of a smoke-emitting chimney should be 100 metres.

State if the height of the above-mentioned chimney meets the pollution norms.

What value is discussed in this question?

From the top of a 7-metre-high building,

the angle of elevation of the top of a cable tower is 60° and the angle of

depression of its foot is 45°. Determine the height of the tower.

The angle of depression from the top of a

tower of a point A on the ground is 30°. On moving a distance of 20 metres from the point A towards the foot of the tower to

a point B, the angle of elevation of the top of the tower from the point B is

60°. Find the height of the tower and its distance from the point A.

The angle of elevation of the top of a

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.

A man on the deck of a ship, 16m above water level observes that the angles of elevation and depression respectively of the top and bottom of a cliff are 60 and 30. Calculate the distance of the cliff from the ship and height of the cliff. _{}

Let AB be the height of the deck and let CD be the cliff..

Let the man be at B, then, AB= 16 m

Let BE _{}CD and AE _{}CD

Then, _{}EBD = 60 and _{}EBC = 30

CE = AB = 16m

Let CD = h meters

Then, ED = (h 16)m

From right _{}BED, we have

_{}

From right _{}CAB, we have

_{}

Hence the height of cliff is 64 m and the distance between the cliff and the ship = _{}

The angle of elevation of the top Q of a

vertical tower PQ from a point X on the ground is 60°. At a point Y, 40 m

vertically above X, the angle of elevation is 45°. Find the height of tower

PQ.

The angle of elevation of an aeroplane from a point on the ground is 45°. After flying

for 15 seconds, the elevation changes to 30°. If the aeroplane

is flying at a height of 2500 metres, find the

speed of the aeroplane.

The angle of elevation of the top of a tower from a point on the same level as the foot of the tower is 30°. On advancing 150 m towards the foot of the tower, the angle of elevation becomes 60°. Show that the height of the tower is 129.9 metres. _{}

Let AB be the tower and let the angle of elevation of its top at C be 30°. Let D be a point at a distance 150 m from C such that the angle of elevation of the top of tower at D is 60°.

Let h m be the height of the tower and AD = x m

In _{}CAB, we have

Hence the height of tower is 129.9 m

As observed from the top of a lighthouse, 100m above sea level, the angle of depression of a ship, sailing directly towards it, changes from 30° to 60°. Determine the distance travelled by the ship during the period of observation. _{}

Let AB be the light house and let C and D be the positions of the ship.

Llet AD =x, CD = y

In _{}BDA,

_{}

The distance travelled by the ship during the period of observation = 115.46 m

From a point on a bridge across a river,

the angles of depression of the banks on opposite sides of the river are 30°

and 45° respectively. If the bridge is at a height of 2.5 m from the banks,

find the width of the river.

The angles of elevation of the top of a

tower from two points at distances of 4 m and 9 m from the base of the tower

and in the same straight line with it are complementary. Show that the height

of the tower is 6 metres.

A ladder of length 6 metres

makes an angle of 45° with the floor while leaning against one wall of a

room. If the foot of the ladder is kept fixed on the floor and it is made to

lean against the opposite wall of the room, it makes an angle of 60° with the

floor. Find the distance between two walls of the room.

From the top of a vertical tower, the

angles of depression of two cars in the same straight line with the base of

the tower, at an instant are found to be 45° and 60°. If the cars are 100 m

apart and are on the same side of the tower, find the height of the tower.

An electrician has to repair an electric

fault on a pole of height 4 metres. He needs to

reach a point 1 metre below the top of the pole to

undertake the repair work. What should be the length of the ladder that he

should use, which when inclined at an angle of 60° to the horizontal would

enable him to reach the required position?

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° and 60° 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.

## Chapter 14 – Height and Distance Exercise MCQ

If the height of a vertical pole is equal to the length of its shadow on the ground, the angle of elevation of the sun is

(a) 0ᵒ

(b) 30ᵒ

(c) 45ᵒ

(d) 60ᵒ

(a) 30ᵒ

(b) 45ᵒ

(c) 60ᵒ

(a) 75ᵒ

(a) 45ᵒ

(b) 30ᵒ

(c) 60ᵒ

(d) 90ᵒ

(a) 60ᵒ

(b) 45ᵒ

(c) 30ᵒ

(b) 90ᵒ

The shadow of a 5-m-long stick is 2 m long. At the same time, the length of the shadow of a 12.5-m-high tree is

(a) 3 m

(b) 3.5 m

(c) 4.5 m

(d) 5 m

A ladder makes an angle of 60ᵒ with the ground when placed against a wall. If the foot of the ladder is 2 m away from the wall, the length of the ladder is

A ladder 15 m long just reaches the top of a vertical wall. If the ladder makes an angle of 60ᵒ with the wall then the height of the wall is

From a point on the ground, 30 m away from the foot of a tower the angle of elevation of the top of the tower is 30°. The height of the tower is

The angle of depression of a car parked on the road from the top of a 150-m-high tower is 30°. The distance of the car from the tower is

A kite is flying at a height of 30 m from the ground. The length of string from the kite to the ground is 60 m. Assuming that there is no slack in the string, the angle of elevation of the kite at the ground is

(a) 45ᵒ

(b) 30ᵒ

(c) 60ᵒ

(a) 90ᵒ

From the top of a cliff 20 m high, the angle of elevation of the top of a tower is found to be equal to the angle of depression of the foot of the tower. The height of the tower is

(a) 20 m

(b) 40 m

(c) 60 m

(d) 80 m

If a 1-5-m-tall girl stands at a distance of 3 m from a lamp post and casts a shadow of length 4.5 m on the ground, then the height of the lamp post is

(a) 1.5 m

(b) 2 m

(c) 2.5 m

(d) 2.8 m

The length of the shadow of a tower standing on level ground is found to be 2x metres longer when the sun’s elevation is 30ᵒ than when it was 45ᵒ. The height of the tower is

(a) 30ᵒ

(b) 45ᵒ

(c) 60ᵒ

(d) 90ᵒ

The tops of two towers of heights x and y, standing on a level ground subtend angles of 30ᵒ and 60ᵒ respectively at the centre of the line joining their feet. Then, x : y is

(a) 1 : 2

(b) 2 : 1

(c) 1 : 3

(d) 3 : 1

The angle of elevation of the top of a tower from a point on the ground 30 m away from the foot of the tower is 30ᵒ. The height of the tower is

The string of a kite is 100 m long and it makes an angle of 60ᵒ with the horizontal. If there is no slack in the string, the height of the kite from the ground is

If the angles of elevation of the top of a tower from two points at distances a and b from the base and in the same straight line with it are complementary then the height of the tower is

On the level ground, the angle of elevation of a tower is 30ᵒ. On moving 20 m nearer, the angle of elevation is 60ᵒ. The height of the tower is

In a rectangle, the angle between a diagonal and a side is 30ᵒ and the length of this diagonal is 8 cm. The area of the rectangle is ,

From the top of a hill, the angles of depression of two consecutive km stones due east are found to be 30ᵒ and 45ᵒ. The height of the hill is

If the elevation of the sun changes from 30ᵒ to 60ᵒ then the difference between the lengths of shadows of a pole 15 m high, is

An observer 1.5 m tall is 28.5 m away from a tower and the angle of elevation of the top of the tower from the eye of the observer is 45ᵒ. The height of the tower is

(a) 27 m

(b) 30 m

(c) 28.5 m

(d) none of these