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

Page / Exercise

Chapter 12 - Some Applications of Trigonometry Exercise 12.41

Question 1

Rd-sharma Solutions Cbse Class 10 Mathematics Chapter - Heights And Distances

Solution 1

Rd-sharma Solutions Cbse Class 10 Mathematics Chapter - Heights And Distances

Rd-sharma Solutions Cbse Class 10 Mathematics Chapter - Heights And Distances

Question 2

Rd-sharma Solutions Cbse Class 10 Mathematics Chapter - Heights And Distances

Solution 2

Rd-sharma Solutions Cbse Class 10 Mathematics Chapter - Heights And Distances

Rd-sharma Solutions Cbse Class 10 Mathematics Chapter - Heights And Distances

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

Rd-sharma Solutions Cbse Class 10 Mathematics Chapter - Heights And Distances

Solution 3

Rd-sharma Solutions Cbse Class 10 Mathematics Chapter - Heights And Distances

Rd-sharma Solutions Cbse Class 10 Mathematics Chapter - Heights And Distances

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

Rd-sharma Solutions Cbse Class 10 Mathematics Chapter - Heights And Distances

Solution 4

Rd-sharma Solutions Cbse Class 10 Mathematics Chapter - Heights And Distances

Rd-sharma Solutions Cbse Class 10 Mathematics Chapter - Heights And Distances

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

Rd-sharma Solutions Cbse Class 10 Mathematics Chapter - Heights And Distances

Solution 5

Rd-sharma Solutions Cbse Class 10 Mathematics Chapter - Heights And Distances

Rd-sharma Solutions Cbse Class 10 Mathematics Chapter - Heights And Distances

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 

Rd-sharma Solutions Cbse Class 10 Mathematics Chapter - Heights And Distances

Solution 6

Rd-sharma Solutions Cbse Class 10 Mathematics Chapter - Heights And Distances

Rd-sharma Solutions Cbse Class 10 Mathematics Chapter - Heights And Distances

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

Rd-sharma Solutions Cbse Class 10 Mathematics Chapter - Heights And Distances

Solution 7

Rd-sharma Solutions Cbse Class 10 Mathematics Chapter - Heights And Distances

Rd-sharma Solutions Cbse Class 10 Mathematics Chapter - Heights And Distances

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

Rd-sharma Solutions Cbse Class 10 Mathematics Chapter - Heights And Distances

Wire BD

ED || AC

So, EA = DC and ED = AC

EA = 14

AB = EA + EB

20 = 14 + EB

EB = 6

Rd-sharma Solutions Cbse Class 10 Mathematics Chapter - Heights And Distances

 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

Rd-sharma Solutions Cbse Class 10 Mathematics Chapter - Heights And Distances

Rd-sharma Solutions Cbse Class 10 Mathematics Chapter - Heights And Distances

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

Rd-sharma Solutions Cbse Class 10 Mathematics Chapter - Heights And Distances

Solution 10

Rd-sharma Solutions Cbse Class 10 Mathematics Chapter - Heights And Distances

Rd-sharma Solutions Cbse Class 10 Mathematics Chapter - Heights And Distances

Chapter 12 - Some Applications of Trigonometry Exercise 12.42

Question 1

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 1

Rd-sharma Solutions Cbse Class 10 Mathematics Chapter - Heights And Distances

Rd-sharma Solutions Cbse Class 10 Mathematics Chapter - Heights And Distances

Question 2

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

Rd-sharma Solutions Cbse Class 10 Mathematics Chapter - Heights And Distances

Solution 2

Rd-sharma Solutions Cbse Class 10 Mathematics Chapter - Heights And Distances

Rd-sharma Solutions Cbse Class 10 Mathematics Chapter - Heights And Distances

Question 3

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

Rd-sharma Solutions Cbse Class 10 Mathematics Chapter - Heights And Distances

Solution 3

Rd-sharma Solutions Cbse Class 10 Mathematics Chapter - Heights And Distances

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.

Rd-sharma Solutions Cbse Class 10 Mathematics Chapter - Heights And Distances

So, the correct option is (d).

Question 4

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 4

Rd-sharma Solutions Cbse Class 10 Mathematics Chapter - Heights And Distances

Rd-sharma Solutions Cbse Class 10 Mathematics Chapter - Heights And Distances

Question 5

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

Rd-sharma Solutions Cbse Class 10 Mathematics Chapter - Heights And Distances

Solution 5

Rd-sharma Solutions Cbse Class 10 Mathematics Chapter - Heights And Distances

Rd-sharma Solutions Cbse Class 10 Mathematics Chapter - Heights And Distances

Question 6

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

Rd-sharma Solutions Cbse Class 10 Mathematics Chapter - Heights And Distances

Solution 6

Rd-sharma Solutions Cbse Class 10 Mathematics Chapter - Heights And Distances

Rd-sharma Solutions Cbse Class 10 Mathematics Chapter - Heights And Distances

Question 7

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

Rd-sharma Solutions Cbse Class 10 Mathematics Chapter - Heights And Distances

Solution 7

Rd-sharma Solutions Cbse Class 10 Mathematics Chapter - Heights And Distances

Rd-sharma Solutions Cbse Class 10 Mathematics Chapter - Heights And Distances

Question 8

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

Rd-sharma Solutions Cbse Class 10 Mathematics Chapter - Heights And Distances

Solution 8

Rd-sharma Solutions Cbse Class 10 Mathematics Chapter - Heights And Distances

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.

Rd-sharma Solutions Cbse Class 10 Mathematics Chapter - Heights And Distances

So, the correct option is (b).

Question 9

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 9

Rd-sharma Solutions Cbse Class 10 Mathematics Chapter - Heights And Distances

EC || AB

Hence

EA = CB = 10

AD = AE + ED

ED = AD - AE

     = 16 - 10 = 6

 Rd-sharma Solutions Cbse Class 10 Mathematics Chapter - Heights And Distances

So, the correct option is (c).

Question 10

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 10

Rd-sharma Solutions Cbse Class 10 Mathematics Chapter - Heights And Distances

Rd-sharma Solutions Cbse Class 10 Mathematics Chapter - Heights And Distances

Chapter 12 - Some Applications of Trigonometry Exercise 12.43

Question 1

Rd-sharma Solutions Cbse Class 10 Mathematics Chapter - Heights And Distances

Solution 1

Rd-sharma Solutions Cbse Class 10 Mathematics Chapter - Heights And Distances

Rd-sharma Solutions Cbse Class 10 Mathematics Chapter - Heights And Distances

Question 2

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

Rd-sharma Solutions Cbse Class 10 Mathematics Chapter - Heights And Distances

Solution 2

Rd-sharma Solutions Cbse Class 10 Mathematics Chapter - Heights And Distances

 

 

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

Rd-sharma Solutions Cbse Class 10 Mathematics Chapter - Heights And Distances

Question 3

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

Rd-sharma Solutions Cbse Class 10 Mathematics Chapter - Heights And Distances

Solution 3

Rd-sharma Solutions Cbse Class 10 Mathematics Chapter - Heights And Distances

Rd-sharma Solutions Cbse Class 10 Mathematics Chapter - Heights And Distances

Question 4

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

Rd-sharma Solutions Cbse Class 10 Mathematics Chapter - Heights And Distances

Solution 4

Rd-sharma Solutions Cbse Class 10 Mathematics Chapter - Heights And Distances

Rd-sharma Solutions Cbse Class 10 Mathematics Chapter - Heights And Distances

Question 5

Rd-sharma Solutions Cbse Class 10 Mathematics Chapter - Heights And Distances

Solution 5

Rd-sharma Solutions Cbse Class 10 Mathematics Chapter - Heights And Distances

Rd-sharma Solutions Cbse Class 10 Mathematics Chapter - Heights And Distances

Question 6

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

Rd-sharma Solutions Cbse Class 10 Mathematics Chapter - Heights And Distances

Solution 6

Rd-sharma Solutions Cbse Class 10 Mathematics Chapter - Heights And Distances

Rd-sharma Solutions Cbse Class 10 Mathematics Chapter - Heights And Distances

Question 7

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

Rd-sharma Solutions Cbse Class 10 Mathematics Chapter - Heights And Distances

Solution 7

Rd-sharma Solutions Cbse Class 10 Mathematics Chapter - Heights And Distances

Rd-sharma Solutions Cbse Class 10 Mathematics Chapter - Heights And Distances

Chapter 12 - Some Applications of Trigonometry Exercise Ex. 12.1

Question 1

Rd-sharma Solutions Cbse Class 10 Mathematics Chapter - Heights And Distances

Solution 1

Rd-sharma Solutions Cbse Class 10 Mathematics Chapter - Heights And Distances

Rd-sharma Solutions Cbse Class 10 Mathematics Chapter - Heights And Distances
Question 2

Rd-sharma Solutions Cbse Class 10 Mathematics Chapter - Heights And Distances

Solution 2

Rd-sharma Solutions Cbse Class 10 Mathematics Chapter - Heights And Distances

Rd-sharma Solutions Cbse Class 10 Mathematics Chapter - Heights And Distances
Question 3

Rd-sharma Solutions Cbse Class 10 Mathematics Chapter - Heights And Distances

Solution 3

Rd-sharma Solutions Cbse Class 10 Mathematics Chapter - Heights And Distances

Rd-sharma Solutions Cbse Class 10 Mathematics Chapter - Heights And Distances
Question 4

Rd-sharma Solutions Cbse Class 10 Mathematics Chapter - Heights And Distances

Solution 4
Rd-sharma Solutions Cbse Class 10 Mathematics Chapter - Heights And Distances



Rd-sharma Solutions Cbse Class 10 Mathematics Chapter - Heights And Distances

Question 5

Rd-sharma Solutions Cbse Class 10 Mathematics Chapter - Heights And Distances

Solution 5

Rd-sharma Solutions Cbse Class 10 Mathematics Chapter - Heights And Distances

Rd-sharma Solutions Cbse Class 10 Mathematics Chapter - Heights And Distances
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

Rd-sharma Solutions Cbse Class 10 Mathematics Chapter - Heights And Distances

Rd-sharma Solutions Cbse Class 10 Mathematics Chapter - Heights And Distances

Question 7
Rd-sharma Solutions Cbse Class 10 Mathematics Chapter - Heights And Distances
Solution 7
Rd-sharma Solutions Cbse Class 10 Mathematics Chapter - Heights And Distances
Rd-sharma Solutions Cbse Class 10 Mathematics Chapter - Heights And Distances
Question 8
Rd-sharma Solutions Cbse Class 10 Mathematics Chapter - Heights And Distances
Solution 8
Rd-sharma Solutions Cbse Class 10 Mathematics Chapter - Heights And Distances


Rd-sharma Solutions Cbse Class 10 Mathematics Chapter - Heights And Distances
Question 9
Rd-sharma Solutions Cbse Class 10 Mathematics Chapter - Heights And Distances
Solution 9
Rd-sharma Solutions Cbse Class 10 Mathematics Chapter - Heights And Distances
Rd-sharma Solutions Cbse Class 10 Mathematics Chapter - Heights And Distances
Question 10
Rd-sharma Solutions Cbse Class 10 Mathematics Chapter - Heights And Distances
Solution 10
Rd-sharma Solutions Cbse Class 10 Mathematics Chapter - Heights And Distances
Rd-sharma Solutions Cbse Class 10 Mathematics Chapter - Heights And Distances
Question 11
Rd-sharma Solutions Cbse Class 10 Mathematics Chapter - Heights And Distances
Solution 11
Rd-sharma Solutions Cbse Class 10 Mathematics Chapter - Heights And Distances
Rd-sharma Solutions Cbse Class 10 Mathematics Chapter - Heights And Distances
Question 12
Rd-sharma Solutions Cbse Class 10 Mathematics Chapter - Heights And Distances
Solution 12
Rd-sharma Solutions Cbse Class 10 Mathematics Chapter - Heights And Distances
Rd-sharma Solutions Cbse Class 10 Mathematics Chapter - Heights And Distances
Question 13
Rd-sharma Solutions Cbse Class 10 Mathematics Chapter - Heights And Distances
Solution 13
Rd-sharma Solutions Cbse Class 10 Mathematics Chapter - Heights And Distances
Rd-sharma Solutions Cbse Class 10 Mathematics Chapter - Heights And Distances
Question 14
Rd-sharma Solutions Cbse Class 10 Mathematics Chapter - Heights And Distances
Solution 14
Rd-sharma Solutions Cbse Class 10 Mathematics Chapter - Heights And Distances
Rd-sharma Solutions Cbse Class 10 Mathematics Chapter - Heights And Distances
Question 15
Rd-sharma Solutions Cbse Class 10 Mathematics Chapter - Heights And Distances
Solution 15
Rd-sharma Solutions Cbse Class 10 Mathematics Chapter - Heights And Distances
Rd-sharma Solutions Cbse Class 10 Mathematics Chapter - Heights And Distances
Question 16
Rd-sharma Solutions Cbse Class 10 Mathematics Chapter - Heights And Distances
Solution 16
Rd-sharma Solutions Cbse Class 10 Mathematics Chapter - Heights And Distances
Rd-sharma Solutions Cbse Class 10 Mathematics Chapter - Heights And Distances
Question 17
Rd-sharma Solutions Cbse Class 10 Mathematics Chapter - Heights And Distances
Solution 17
Rd-sharma Solutions Cbse Class 10 Mathematics Chapter - Heights And Distances
Rd-sharma Solutions Cbse Class 10 Mathematics Chapter - Heights And Distances
Question 18
Rd-sharma Solutions Cbse Class 10 Mathematics Chapter - Heights And Distances
Solution 18
Rd-sharma Solutions Cbse Class 10 Mathematics Chapter - Heights And Distances
Rd-sharma Solutions Cbse Class 10 Mathematics Chapter - Heights And Distances
Question 19
Rd-sharma Solutions Cbse Class 10 Mathematics Chapter - Heights And Distances
Solution 19
Rd-sharma Solutions Cbse Class 10 Mathematics Chapter - Heights And Distances
Rd-sharma Solutions Cbse Class 10 Mathematics Chapter - Heights And Distances
Question 20
Rd-sharma Solutions Cbse Class 10 Mathematics Chapter - Heights And Distances
Solution 20
Rd-sharma Solutions Cbse Class 10 Mathematics Chapter - Heights And Distances
Rd-sharma Solutions Cbse Class 10 Mathematics Chapter - Heights And Distances
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
Rd-sharma Solutions Cbse Class 10 Mathematics Chapter - Heights And Distances

Rd-sharma Solutions Cbse Class 10 Mathematics Chapter - Heights And Distances

Rd-sharma Solutions Cbse Class 10 Mathematics Chapter - Heights And Distances
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
Rd-sharma Solutions Cbse Class 10 Mathematics Chapter - Heights And Distances
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
Rd-sharma Solutions Cbse Class 10 Mathematics Chapter - Heights And Distances
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
Rd-sharma Solutions Cbse Class 10 Mathematics Chapter - Heights And Distances

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.

Rd-sharma Solutions Cbse Class 10 Mathematics Chapter - Heights And Distances
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
Rd-sharma Solutions Cbse Class 10 Mathematics Chapter - Heights And Distances
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
Rd-sharma Solutions Cbse Class 10 Mathematics Chapter - Heights And Distances

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

Rd-sharma Solutions Cbse Class 10 Mathematics Chapter - Heights And Distances
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
Rd-sharma Solutions Cbse Class 10 Mathematics Chapter - Heights And Distances

Rd-sharma Solutions Cbse Class 10 Mathematics Chapter - Heights And Distances
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
Rd-sharma Solutions Cbse Class 10 Mathematics Chapter - Heights And Distances
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
Rd-sharma Solutions Cbse Class 10 Mathematics Chapter - Heights And Distances

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

Rd-sharma Solutions Cbse Class 10 Mathematics Chapter - Heights And Distances
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
Rd-sharma Solutions Cbse Class 10 Mathematics Chapter - Heights And Distances
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
Rd-sharma Solutions Cbse Class 10 Mathematics Chapter - Heights And Distances
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
Rd-sharma Solutions Cbse Class 10 Mathematics Chapter - Heights And Distances
Question 33
Rd-sharma Solutions Cbse Class 10 Mathematics Chapter - Heights And Distances
Solution 33
Rd-sharma Solutions Cbse Class 10 Mathematics Chapter - Heights And Distances

Rd-sharma Solutions Cbse Class 10 Mathematics Chapter - Heights And Distances
Question 34
Rd-sharma Solutions Cbse Class 10 Mathematics Chapter - Heights And Distances
Solution 34

Rd-sharma Solutions Cbse Class 10 Mathematics Chapter - Heights And Distances
Rd-sharma Solutions Cbse Class 10 Mathematics Chapter - Heights And Distances
Question 35
Rd-sharma Solutions Cbse Class 10 Mathematics Chapter - Heights And Distances
Solution 35
Rd-sharma Solutions Cbse Class 10 Mathematics Chapter - Heights And Distances

Rd-sharma Solutions Cbse Class 10 Mathematics Chapter - Heights And Distances
Question 36
Rd-sharma Solutions Cbse Class 10 Mathematics Chapter - Heights And Distances
Solution 36
Rd-sharma Solutions Cbse Class 10 Mathematics Chapter - Heights And Distances

Rd-sharma Solutions Cbse Class 10 Mathematics Chapter - Heights And Distances
Question 37
Rd-sharma Solutions Cbse Class 10 Mathematics Chapter - Heights And Distances
Solution 37
Rd-sharma Solutions Cbse Class 10 Mathematics Chapter - Heights And Distances

Rd-sharma Solutions Cbse Class 10 Mathematics Chapter - Heights And Distances
Question 38
Rd-sharma Solutions Cbse Class 10 Mathematics Chapter - Heights And Distances
Solution 38
Rd-sharma Solutions Cbse Class 10 Mathematics Chapter - Heights And Distances

Rd-sharma Solutions Cbse Class 10 Mathematics Chapter - Heights And Distances
Question 39
Rd-sharma Solutions Cbse Class 10 Mathematics Chapter - Heights And Distances
Solution 39
Rd-sharma Solutions Cbse Class 10 Mathematics Chapter - Heights And Distances

Rd-sharma Solutions Cbse Class 10 Mathematics Chapter - Heights And Distances
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.

 

Rd-sharma Solutions Cbse Class 10 Mathematics Chapter - Heights And Distances

Solution 40

 

Rd-sharma Solutions Cbse Class 10 Mathematics Chapter - Heights And Distances

 

Rd-sharma Solutions Cbse Class 10 Mathematics Chapter - Heights And Distances 

 

 

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

Rd-sharma Solutions Cbse Class 10 Mathematics Chapter - Heights And Distances

 

 

Let AC = h be the height of the chimney.

Height of the tower = DE = BC = 40 m

 

In ABE,

Rd-sharma Solutions Cbse Class 10 Mathematics Chapter - Heights And Distances

 Rd-sharma Solutions Cbse Class 10 Mathematics Chapter - Heights And Distances

AB = BE√3….(i)

 

In ∆CBE,

tan 30° = Rd-sharma Solutions Cbse Class 10 Mathematics Chapter - Heights And Distances

Rd-sharma Solutions Cbse Class 10 Mathematics Chapter - Heights And Distances

 

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.

 

Rd-sharma Solutions Cbse Class 10 Mathematics Chapter - Heights And Distances

Solution 42

Rd-sharma Solutions Cbse Class 10 Mathematics Chapter - Heights And Distances

Let the ships be at B and C.

 

In D ABD,

 

Rd-sharma Solutions Cbse Class 10 Mathematics Chapter - Heights And Distances

BD = 200 m

 

In D ADC,

 

 

Rd-sharma Solutions Cbse Class 10 Mathematics Chapter - Heights And Distances

Distance between the two ships = BC = BD + DC

Rd-sharma Solutions Cbse Class 10 Mathematics Chapter - Heights And Distances

Rd-sharma Solutions Cbse Class 10 Mathematics Chapter - Heights And Distances

 

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.

Rd-sharma Solutions Cbse Class 10 Mathematics Chapter - Heights And Distances

Solution 43

Rd-sharma Solutions Cbse Class 10 Mathematics Chapter - Heights And Distances

Here mCAB = mFEB = 30°.

Let BC = h m, AC = x m

In D ADE,

 

Rd-sharma Solutions Cbse Class 10 Mathematics Chapter - Heights And Distances

 

In D BAC,

 

Rd-sharma Solutions Cbse Class 10 Mathematics Chapter - Heights And Distances

 

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

Rd-sharma Solutions Cbse Class 10 Mathematics Chapter - Heights And Distances

Rd-sharma Solutions Cbse Class 10 Mathematics Chapter - Heights And Distances

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

Rd-sharma Solutions Cbse Class 10 Mathematics Chapter - Heights And Distances

Let AQ be the tower and R, S respectively be the points which are 4m, 9m away from base of tower.
Rd-sharma Solutions Cbse Class 10 Mathematics Chapter - Heights And Distances

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

Question 46
Rd-sharma Solutions Cbse Class 10 Mathematics Chapter - Heights And Distances
Solution 46
Rd-sharma Solutions Cbse Class 10 Mathematics Chapter - Heights And Distances
Rd-sharma Solutions Cbse Class 10 Mathematics Chapter - Heights And Distances
Question 47

Rd-sharma Solutions Cbse Class 10 Mathematics Chapter - Heights And Distances

Solution 47

Rd-sharma Solutions Cbse Class 10 Mathematics Chapter - Heights And Distances

Rd-sharma Solutions Cbse Class 10 Mathematics Chapter - Heights And Distances

Question 48

Rd-sharma Solutions Cbse Class 10 Mathematics Chapter - Heights And Distances

Solution 48

Rd-sharma Solutions Cbse Class 10 Mathematics Chapter - Heights And Distances

Rd-sharma Solutions Cbse Class 10 Mathematics Chapter - Heights And Distances
Question 49
Rd-sharma Solutions Cbse Class 10 Mathematics Chapter - Heights And Distances
Solution 49
Rd-sharma Solutions Cbse Class 10 Mathematics Chapter - Heights And Distances


Rd-sharma Solutions Cbse Class 10 Mathematics Chapter - Heights And Distances
Question 50

Rd-sharma Solutions Cbse Class 10 Mathematics Chapter - Heights And Distances

Solution 50

Rd-sharma Solutions Cbse Class 10 Mathematics Chapter - Heights And Distances

Rd-sharma Solutions Cbse Class 10 Mathematics Chapter - Heights And Distances

Question 51

Rd-sharma Solutions Cbse Class 10 Mathematics Chapter - Heights And Distances

Solution 51

Rd-sharma Solutions Cbse Class 10 Mathematics Chapter - Heights And Distances


Rd-sharma Solutions Cbse Class 10 Mathematics Chapter - Heights And Distances

Question 52

Rd-sharma Solutions Cbse Class 10 Mathematics Chapter - Heights And Distances

Solution 52

Rd-sharma Solutions Cbse Class 10 Mathematics Chapter - Heights And Distances

Rd-sharma Solutions Cbse Class 10 Mathematics Chapter - Heights And Distances

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

Rd-sharma Solutions Cbse Class 10 Mathematics Chapter - Heights And Distances


Rd-sharma Solutions Cbse Class 10 Mathematics Chapter - Heights And Distances

Question 54

Rd-sharma Solutions Cbse Class 10 Mathematics Chapter - Heights And Distances

Solution 54

Rd-sharma Solutions Cbse Class 10 Mathematics Chapter - Heights And Distances


Rd-sharma Solutions Cbse Class 10 Mathematics Chapter - Heights And Distances

Question 55

Rd-sharma Solutions Cbse Class 10 Mathematics Chapter - Heights And Distances

Solution 55

Rd-sharma Solutions Cbse Class 10 Mathematics Chapter - Heights And Distances

Rd-sharma Solutions Cbse Class 10 Mathematics Chapter - Heights And Distances

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

Rd-sharma Solutions Cbse Class 10 Mathematics Chapter - Heights And Distances 

 

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)= Rd-sharma Solutions Cbse Class 10 Mathematics Chapter - Heights And Distances

Rd-sharma Solutions Cbse Class 10 Mathematics Chapter - Heights And Distances

In ΔABD,

tan(ADB)= Rd-sharma Solutions Cbse Class 10 Mathematics Chapter - Heights And Distances

Rd-sharma Solutions Cbse Class 10 Mathematics Chapter - Heights And Distances

So, DC=BD - BC

  =Rd-sharma Solutions Cbse Class 10 Mathematics Chapter - Heights And Distances

Now,

Rd-sharma Solutions Cbse Class 10 Mathematics Chapter - Heights And Distances

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. Rd-sharma Solutions Cbse Class 10 Mathematics Chapter - Heights And Distances

Solution 57

Rd-sharma Solutions Cbse Class 10 Mathematics Chapter - Heights And Distances 

 

AB is the tower.

DC is the distance between cars.

AB=120m

In ΔABC,

tan(ACB) = Rd-sharma Solutions Cbse Class 10 Mathematics Chapter - Heights And Distances

Rd-sharma Solutions Cbse Class 10 Mathematics Chapter - Heights And Distances

In ΔABD,

tan(ADB) = Rd-sharma Solutions Cbse Class 10 Mathematics Chapter - Heights And Distances

Rd-sharma Solutions Cbse Class 10 Mathematics Chapter - Heights And Distances

So, DC=BD+BC

Rd-sharma Solutions Cbse Class 10 Mathematics Chapter - Heights And Distances

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

Rd-sharma Solutions Cbse Class 10 Mathematics Chapter - Heights And Distances 

Let CD be the tower.

So CD =15m

AB is the distance between the points.

CAD = 60o and CBD = 45o

ADC = 90o

In ΔADC,

tan(CAD)= Rd-sharma Solutions Cbse Class 10 Mathematics Chapter - Heights And Distances

Rd-sharma Solutions Cbse Class 10 Mathematics Chapter - Heights And Distances

In ΔCBD,

tan(CBD)= Rd-sharma Solutions Cbse Class 10 Mathematics Chapter - Heights And Distances

Rd-sharma Solutions Cbse Class 10 Mathematics Chapter - Heights And Distances

So AB=BD - AD

Rd-sharma Solutions Cbse Class 10 Mathematics Chapter - Heights And Distances

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

Rd-sharma Solutions Cbse Class 10 Mathematics Chapter - Heights And Distances

Rd-sharma Solutions Cbse Class 10 Mathematics Chapter - Heights And Distances
Now, in triangle APB,

sin 60o = AB/ BP

√3/2 = h/ BP

This gives

h = 14.64 km

Question 60

Rd-sharma Solutions Cbse Class 10 Mathematics Chapter - Heights And Distances

Solution 60

Rd-sharma Solutions Cbse Class 10 Mathematics Chapter - Heights And Distances

Rd-sharma Solutions Cbse Class 10 Mathematics Chapter - Heights And Distances

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

Rd-sharma Solutions Cbse Class 10 Mathematics Chapter - Heights And Distances

Rd-sharma Solutions Cbse Class 10 Mathematics Chapter - Heights And Distances
Question 62

Rd-sharma Solutions Cbse Class 10 Mathematics Chapter - Heights And Distances

Solution 62

Rd-sharma Solutions Cbse Class 10 Mathematics Chapter - Heights And Distances

Rd-sharma Solutions Cbse Class 10 Mathematics Chapter - Heights And Distances
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

Rd-sharma Solutions Cbse Class 10 Mathematics Chapter - Heights And Distances

Rd-sharma Solutions Cbse Class 10 Mathematics Chapter - Heights And Distances

Question 64

Rd-sharma Solutions Cbse Class 10 Mathematics Chapter - Heights And Distances

Solution 64

Rd-sharma Solutions Cbse Class 10 Mathematics Chapter - Heights And Distances

Rd-sharma Solutions Cbse Class 10 Mathematics Chapter - Heights And Distances

Question 65

Rd-sharma Solutions Cbse Class 10 Mathematics Chapter - Heights And Distances

Solution 65

Rd-sharma Solutions Cbse Class 10 Mathematics Chapter - Heights And Distances

Rd-sharma Solutions Cbse Class 10 Mathematics Chapter - Heights And Distances
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

Rd-sharma Solutions Cbse Class 10 Mathematics Chapter - Heights And Distances
Rd-sharma Solutions Cbse Class 10 Mathematics Chapter - Heights And Distances
Question 67

Rd-sharma Solutions Cbse Class 10 Mathematics Chapter - Heights And Distances

Solution 67

Rd-sharma Solutions Cbse Class 10 Mathematics Chapter - Heights And Distances

Rd-sharma Solutions Cbse Class 10 Mathematics Chapter - Heights And Distances

Question 68

Rd-sharma Solutions Cbse Class 10 Mathematics Chapter - Heights And Distances

Solution 68

Rd-sharma Solutions Cbse Class 10 Mathematics Chapter - Heights And Distances

Rd-sharma Solutions Cbse Class 10 Mathematics Chapter - Heights And Distances

Question 69

Rd-sharma Solutions Cbse Class 10 Mathematics Chapter - Heights And Distances

Solution 69

Rd-sharma Solutions Cbse Class 10 Mathematics Chapter - Heights And Distances


Rd-sharma Solutions Cbse Class 10 Mathematics Chapter - Heights And Distances

Question 70

Rd-sharma Solutions Cbse Class 10 Mathematics Chapter - Heights And Distances

Solution 70

Rd-sharma Solutions Cbse Class 10 Mathematics Chapter - Heights And Distances

Rd-sharma Solutions Cbse Class 10 Mathematics Chapter - Heights And Distances

Question 71

Rd-sharma Solutions Cbse Class 10 Mathematics Chapter - Heights And Distances

Solution 71

Rd-sharma Solutions Cbse Class 10 Mathematics Chapter - Heights And Distances

Rd-sharma Solutions Cbse Class 10 Mathematics Chapter - Heights And Distances

Question 72

Rd-sharma Solutions Cbse Class 10 Mathematics Chapter - Heights And Distances

Solution 72

Rd-sharma Solutions Cbse Class 10 Mathematics Chapter - Heights And Distances

Rd-sharma Solutions Cbse Class 10 Mathematics Chapter - Heights And Distances

Question 73

Rd-sharma Solutions Cbse Class 10 Mathematics Chapter - Heights And Distances

Solution 73

Rd-sharma Solutions Cbse Class 10 Mathematics Chapter - Heights And Distances

Rd-sharma Solutions Cbse Class 10 Mathematics Chapter - Heights And Distances

Question 74

Rd-sharma Solutions Cbse Class 10 Mathematics Chapter - Heights And Distances

Solution 74

Rd-sharma Solutions Cbse Class 10 Mathematics Chapter - Heights And Distances

Rd-sharma Solutions Cbse Class 10 Mathematics Chapter - Heights And Distances
Question 75

Rd-sharma Solutions Cbse Class 10 Mathematics Chapter - Heights And Distances

Solution 75

Rd-sharma Solutions Cbse Class 10 Mathematics Chapter - Heights And Distances

Rd-sharma Solutions Cbse Class 10 Mathematics Chapter - Heights And Distances

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

Rd-sharma Solutions Cbse Class 10 Mathematics Chapter - Heights And Distances

Rd-sharma Solutions Cbse Class 10 Mathematics Chapter - Heights And Distances

Rd-sharma Solutions Cbse Class 10 Mathematics Chapter - Heights And Distances

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

Rd-sharma Solutions Cbse Class 10 Mathematics Chapter - Heights And Distances

Rd-sharma Solutions Cbse Class 10 Mathematics Chapter - Heights And Distances

 

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

Rd-sharma Solutions Cbse Class 10 Mathematics Chapter - Heights And Distances

 

Rd-sharma Solutions Cbse Class 10 Mathematics Chapter - Heights And Distances