Table of Contents
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 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 30o to 60o 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 30o. 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 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 25The 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 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 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 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 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 28From 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 29As 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 30The 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 31From 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 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 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
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 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 (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 = 60o and ∠ADB = 45o
∠ABC = 90o
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 60o to 30o. 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 = 60o and ∠CBD = 45o
∠ADC = 90o
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 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 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 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
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