# RD SHARMA Solutions for Class 10 Maths Chapter 12 - Heights and Distances

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## Chapter 12 - Some Applications of Trigonometry Exercise Ex. 12.1

Question 1

Solution 1

Question 2

Solution 2

Question 3

Solution 3

Question 4

Solution 4

Question 5

Solution 5

Question 6

A ladder 15 metres long just reaches the top of a vertical wall. If the ladder makes an angle of 60° with the wall, find the height of the wall.

Solution 6

Question 7
Solution 7

Question 8
Solution 8

Question 9
Solution 9
Question 10
Solution 10
Question 11
Solution 11
Question 12
Solution 12
Question 13
Solution 13
Question 14
Solution 14
Question 15
Solution 15
Question 16
Solution 16
Question 17
Solution 17
Question 18
Solution 18
Question 19
Solution 19
Question 20
Solution 20
Question 21
A 1.6 m tall girl stands at a distance of 3.2 m from a lamp-post and casts a shadow of 4.8 m on the ground. Find the height of the lamp-post by using (i) trigonometric ratios (ii) property of similar triangles.
Solution 21

Question 22
A 1.5 m tall boy is standing at some distance from a 30 m tall building. The angle of elevation from his eyes to the top of the building increases from 30o to 60o as he walks towards the building. Find the distance he walked towards the building.
Solution 22
Question 23
The shadow of a tower standing on a level ground is found to be 40 m longer when Sun'saltitude is 30o. Find the height of the tower.
Solution 23

Question 24
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 45o and 60o respectively. Find the height of the tower.
Solution 24

Let BC be the building, AB be the transmission tower, and D be the point on ground from where elevation angles are to be measured.

Question 25
The angles of depression of the top and bottom of 8 m tall building from the top of a multistoried building are 30o and 45o respectively. Find the height of the multistoried building and the distance between the two buildings.
Solution 25

Question 26
A statue, 1.6 m tall, stands on a top of pedestal, from a point on the ground, the angle of elevation of the top of statue is 60o and from the same point the angle of elevation of the top of the pedestal is 45o. Find the height of the pedestal.
Solution 26

Let AB be the statue, BC be the pedestal and D be the point on ground from where elevation angles are to be measured.

Question 27
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 60o. 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 30o. Find the height of the tower and the width of the canal.

Solution 27

Question 28
From the top of a 7 m high building, the angle of elevation of the top of a cable tower is 60o and the angle of depression of its foot is 45o. Determine the height of the tower.
Solution 28
Question 29
As observed from the top of a 75 m high lighthouse from the sea-level, the angles of depression of two ships are 30o and 45o. If one ship is exactly behind the other on the same side of the lighthouse, find the distance between the two ships.
Solution 29

Let AB be the lighthouse and the two ships be at point C and D respectively.

Question 30
The angle of elevation of the top of a building from the foot of the tower is 30o and the angle of elevation of the top of the tower from the foot of the building is 60o. If the tower is 50 m high, find the height of the building.
Solution 30
Question 31
From a point on a bridge across a river the angles of depression of the banks on opposite side of the river are 30o and 45o respectively. If bridge is at the height of 30 m from the banks, find the width of the river.
Solution 31

Question 32
Two poles of equal heights are standing opposite each other an either side of the road, which is 80 m wide. From a point between them on the road, the angles of elevation of the top of the poles are 60o and 30o, respectively. Find the height of poles and the distance of the point from the poles.
Solution 32
Question 33
Solution 33

Question 34
Solution 34

Question 35
Solution 35

Question 36
Solution 36

Question 37
Solution 37

Question 38
Solution 38

Question 39
Solution 39

Question 40

An aeroplane is flying at a height of 210 m. Flying at this height at some instant the angles of depression of two points in a line in opposite directions on both the banks of the river are 45° and 60°. Find the width of the river.

Solution 40

Question 41

The angle of elevation of the top of a chimney from the top of the 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, 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 pollution norms. What value is discussed in this question?

Solution 41

Let AC = h be the height of the chimney.

Height of the tower = DE = BC = 40 m

In ABE,

AB = BE√3….(i)

In ∆CBE,

tan 30° =

Substituting BE in (i),

AB = 40√3 × 3

= 120 m

Height of the chimney = AB + BC = 120 + 40 = 160 m

Yes, the height of the chimney meets the pollution control norms.

Question 42

Two ships are there in the sea on either side of a light house in such away that the ships and the light house are in the same straight line. The angles of depression of two ships are observed from the top of the light house are 60° and 45°  respectively. If the height of the light house is 200 m, find the distance between the two ships.

Solution 42

Let the ships be at B and C.

In D ABD,

BD = 200 m

Distance between the two ships = BC = BD + DC

Question 43

The horizontal distance between two poles is 15 m. The angle of depression of the top of the first pole as seen from the top of the second pole is 30°. If the height of the second pole is 24 m, find the height of the first pole.

Solution 43

Here mCAB = mFEB = 30°.

Let BC = h m, AC = x m

In D BAC,

Height of the second pole is 15.34 m

Question 44

The angles of depression of two ships from the top of a light house and on the same side of it are found to be 45o and 30o respectively. If the ships are 200 m apart, find the height of the light house.

Solution 44

Question 45

The angles of elevation of the top of a tower from two points at a distance of 4 m and 9 m from the base of the tower and in the same straight line with it are complementary. Prove that the height of the tower is 6 m.

Solution 45

Let AQ be the tower and R, S respectively be the points which are 4m, 9m away from base of tower.

As the height can not be negative, the height of the tower is 6 m.

Question 46
Solution 46

Question 47

Solution 47

Question 48

Solution 48

Question 49
Solution 49

Question 50

Solution 50

Question 51

Solution 51

Question 52

Solution 52

Question 53

From the top of building AB, 60 m high, the angles of depression of the top and bottom of a vertical lamp post CD are observed to be 30o and 60o 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.

Solution 53

Question 54

Solution 54

Question 55

Solution 55

Question 56

ΔA moving boat is observed from the top of a 150m high cliff moving away from the cliff. The angle of depression of the boat changes from 60˚ to 45˚ in 2 minutes. Find the speed of the boat in m /h.

Solution 56

Let AB be the cliff, so AB=150m.

C and D are positions of the boat.

DC is the distance covered in 2 min.

ACB = 60o and ADB = 45o

ABC = 90o

In ΔABC,

tan(ACB)=

In ΔABD,

So, DC=BD - BC

=

Now,

Question 57

From the top of a 120 m 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˚ and 45˚. Find the distance between the cars.

Solution 57

AB is the tower.

DC is the distance between cars.

AB=120m

In ΔABC,

tan(ACB) =

In ΔABD,

So, DC=BD+BC

Question 58

Two points A and B are on the same side of a tower and in the same straight line with its base. The angles of depression of these points from the top of the tower are 60˚ and 45˚ respectively. If the height of the tower is 15 m, then find the distance between these points.

Solution 58

Let CD be the tower.

So CD =15m

AB is the distance between the points.

CAD = 60o and CBD = 45o

In ΔCBD,

tan(CBD)=

Question 59

A fire in a building B is reported on telephone to two fire stations P and Q, 20 km apart from each other on a straight road. P observes that the fire is at an angle of 60o to the road and Q observes that it is at an angle of 45o to the road. Which station should send its team and how much will this team have to travel?

Solution 59

Now, in triangle APB,

sin 60o = AB/ BP

√3/2 = h/ BP

This gives

h = 14.64 km

Question 60

Solution 60

Question 61

A man standing on the deck of a ship, which is 8 m above water level. He observes the angle of elevation of the top of a hill as 60o and the angle of depression of the base of the hill as 30o. Calculate the distance of the hill from the ship and the height of the hill.

Solution 61

Question 62

Solution 62

Question 63

The angle of elevation of an aeroplane from a point on the ground is 45o. After a flight of 15 seconds, the elevation changes to 30o. If the aeroplane is flying at a height of 3000 metres, find the speed of the aeroplane.

Solution 63

Question 64

Solution 64

Question 65

Solution 65

Question 66

The angle of elevation of a stationery cloud from a point 2500 m above a lake is 15o and the angle of depression of its reflection in lake is 45o. What is the height of the cloud above the lake level? (Use tan 15o = 0.268)

Solution 66

Question 67

Solution 67

Question 68

Solution 68

Question 69

Solution 69

Question 70

Solution 70

Question 71

Solution 71

Question 72

Solution 72

Question 73

Solution 73

Question 74

Solution 74

Question 75

Solution 75

Question 76

From the top of a tower h metre high, the angles of depression of two objects, which are in the line with the foot of the tower are α and β (β > α). Find the distance between the two objects.

Solution 76

Question 77

A window of a house is h metre above the ground. From the window, the angles of elevation and depression of the top and bottom of another house situated on the opposite side of the lane are found to be a and b respectively. Prove that the height of the house is h (1 + tan α cot β) metres.

Solution 77

Question 78

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 window are observed to be 60° and 30° respectively. Find the height of the balloon above the ground.

Solution 78

## Chapter 12 - Some Applications of Trigonometry Exercise 12.41

Question 1

Solution 1

Question 2

Solution 2

Question 3

If the altitude of the sum is at 60°, then the height of the vertical tower that will cast a shadow of length 30 m is

Solution 3

Question 4

If the angles of elevation of a tower from two points distant a and b (a > b) from its foot and in the same straight line from it are 30° and 60°, then the height of the tower is

Solution 4

Question 5

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

Solution 5

Question 6

From a light house the angles of depression of two ships on opposite sides of the light house are observed to be 30° and 45°. If the height of the light house is h metres, the distance between the ships is

Solution 6

Question 7

The angle of elevation of the top of a tower standing on a horizontal plane from a point A is α. After walking a distance d towards the foot of the tower the angle of elevation is found to be β. The height of the tower is

Solution 7

Question 8

The tops of two poles of height 20 m and 14 m are connected by a wire. If the wire makes an angle of 30° with horizontal, then the length of the wire is

(a) 12 m

(b) 10 m

(c) 8 m

(d) 6 m

Solution 8

Wire BD

ED || AC

So, EA = DC and ED = AC

EA = 14

AB = EA + EB

20 = 14 + EB

EB = 6

So, the correct option is (a).

Question 9

From the top of a cliff 25 m high the angle of elevation 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) 25 m

(b) 50 m

(c) 75 m

(d) 100 m

Solution 9

Question 10

The angles of depression of two ship from the top of a light house are 45° and 30° towards east. If the ships are 100 m apart, the height of the light house is

Solution 10

## Chapter 12 - Some Applications of Trigonometry Exercise 12.42

Question 11

If the angle of elevation of a cloud from point 200 m above a lake is 30° and the angle depression of its reflection in the lake is 60°, then the height of the cloud above the lake, is

(a) 200 m

(b) 500 m

(c) 30 m

(d) 400 m

Solution 11

Question 12

The height of a tower is 100 m. When the angle of elevation of the sun changes from 30° to 45°, the shadow of the tower becomes x metres less. The value of x is

(a) 100 m

Solution 12

Question 13

Two persons are a metres apart and the height of one is double that of the other. If from the middle point of the line joining their feet, an observer finds the angular elevation of their tops to be complementary, then the height of the shorter post is

Solution 13

If height of one person is x then height of another one is 2x. Also If angle of elevation of one is θ then for another it is 90 - θ.

AB = a

C is mid point.

So, the correct option is (d).

Question 14

The angle of elevation of a cloud from a point h metre above a lake is θ. The angle of depression of its reflection in the lake is 45°. The height of the cloud is

(a) h tan (45° + θ)

(b) h cot (45° - θ)

(c) h tan (45° - θ)

(d) h cot (45° + θ)

Solution 14

Question 15

A tower subtends an angle of 30° at a point on the same level as its foot. At a second point h metres above the first, the depression of the foot of the tower is 60°. The height of the tower is

Solution 15

Question 16

It is found that on walking x metres 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

Solution 16

Question 17

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 30° than when it was 45°. The height of the tower in metres is

Solution 17

Question 18

Two poles are 'a' metres apart and the height of one is double of the other . If from the middle point of the line joining their feet an observer finds the angular elevations of their tops to be complementary, then the height of the smaller is

Solution 18

If height of one pole is x then height of the other one is 2x. Also If the angle of elevation of one is θ then for the other it is

90 - θ.

AB = a

C is mid point.

So, the correct option is (b).

Question 19

The tops of two poles of height 16 m and 10 m are connected by a wire of length l metres. If the wire makes an angle 30° with the horizontal, then l =

(a) 26

(b) 16

(c) 12

(d) 10

Solution 19

EC || AB

Hence

EA = CB = 10

= 16 - 10 = 6

So, the correct option is (c).

Question 20

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

Solution 20

## Chapter 12 - Some Applications of Trigonometry Exercise 12.43

Question 21

Solution 21

Question 22

The angle of depression of a car, standing on the ground, from the top of a 75 m tower, is 30°. The distance of the car from the base of the tower (in metres) is

Solution 22

From the figure, it is cleared that we have to find the length of BC.

Question 23

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

Solution 23

Question 24

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 (in metres) is

Solution 24

Question 25

Solution 25

Question 26

The angle of elevation of the top of a tower at a point on the ground 50 m away from the foot of tower is 45°. Then the height of the tower (in metre) is

Solution 26

Question 27

A ladder makes an angle of 60° with the ground when placed against a wall. If the foot of the ladder is 2 m away from the wall, then the length of the ladder (in metres) is

Solution 27