RD SHARMA SOLUTION CHAPTER- 12 Some Applications of Trigonometry| CLASS 10TH MATHEMATICS-EDUGROWN

Chapter 12 – Some Applications of Trigonometry Exercise Ex. 12.1

Question 1

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Question 2

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Question 3

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Question 4

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Question 5

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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

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Question 8

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Question 9

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Question 10

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Question 11

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Question 12

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Question 13

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Question 14

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Question 15

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Question 16

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Question 17

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Question 18

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Question 19

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Question 20

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Question 21A 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 22A 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 30^o to 60^o as he walks towards the building. Find the distance he walked towards the building.Solution 22

Question 23The shadow of a tower standing on a level ground is found to be 40 m longer when Sun’saltitude is 30^o. Find the height of the tower.Solution 23

Question 24From 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.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 25The angles of depression of the top and bottom of 8 m tall building from the top of a multistoried building are 30^o and 45^o respectively. Find the height of the multistoried building and the distance between the two buildings.Solution 25

Question 26A 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 60^o and from the same point the angle of elevation of the top of the pedestal is 45^o. 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 27A 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. Find the height of the tower and the width of the canal.

Solution 27

Question 28From 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.Solution 28

Question 29As observed from the top of a 75 m high lighthouse 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 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 30The angle of elevation of the top of a building from the foot of the 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.Solution 30

Question 31From a point on a bridge across a river the angles of depression of the banks on opposite side of the river are 30^o and 45^o respectively. If bridge is at the height of 30 m from the banks, find the width of the river.Solution 31

Question 32Two 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 60^o and 30^o, respectively. Find the height of poles and the distance of the point from the poles.Solution 32

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Question 34

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Question 35

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Question 36

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Question 37

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Question 38

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Question 39

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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

In D ADC,

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 m∠CAB = m∠FEB = 30°.

Let BC = h m, AC = x m

In D ADE,

In D BAC,

Height of the second pole is 15.34 mQuestion 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 45^o and 30^o 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

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Question 47

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Question 48

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Question 49

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Question 50

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Question 51

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Question 52

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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 30^o and 60^o 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

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Question 55

Solution 55

Question 56 (i)

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 (i)

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 = 60^o and ∠ADB = 45^o

∠ABC = 90^o

In ΔABC,

tan(∠ACB)= 

In ΔABD,

tan(∠ADB)= 

So, DC=BD – BC

  =

Now,

Question 56 (ii)

A man in a boat rowing away from a light house 100 m high takes 2 minutes to change the angle of elevation of the top of the light house from 60^o to 30^o. Find the speed of the boat in metres per minute. (use ) Solution 56 (ii)

Let AB be the lighthouse and C be the position of man initially.

Suppose, a man changes his position from C to D.

As per the question, we obtain the following figure

Let speed of the boat be x metres per minute.

Therefore, CD = 2x

Using trigonometry, we have

Also,

Hence, speed of the boat is 57.8 m.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,

tan(∠ADB) = 

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 = 60^o and ∠CBD = 45^o

∠ADC = 90^o

In ΔADC,

tan(∠CAD)= 

In ΔCBD,

tan(∠CBD)= 

So AB=BD – AD

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 60^o to the road and Q observes that it is at an angle of 45^o 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 60^o = AB/ BP

√3/2 = h/ BP

This gives

h = 14.64 kmQuestion 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 60^o and the angle of depression of the base of the hill as 30^o. Calculate the distance of the hill from the ship and the height of the hill.Solution 61

Question 62

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Question 63

The angle of elevation of an aeroplane from a point on the ground is 45^o. After a flight of 15 seconds, the elevation changes to 30^o. If the aeroplane is flying at a height of 3000 metres, find the speed of the aeroplane.Solution 63

Question 64

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Question 65

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Question 66

The angle of elevation of a stationery cloud from a point 2500 m above a lake is 15^o and the angle of depression of its reflection in lake is 45^o. What is the height of the cloud above the lake level? (Use tan 15^o = 0.268)Solution 66

Question 67

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Question 68

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Question 69

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Question 70

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Question 71

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Question 72

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Question 73

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Question 74

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Question 75

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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