Chapter 9. Some Application of Trigonometry – Chapter-Wise Important Questions and Answers | Class 10 Mathematics CBSE Solutions
CBSE Class 10 Mathematics Chapter 9. Some Application of Trigonometry Important Questions with Solutions and Answers
Chapter 9. SOME APPLICATION OF TRIGONOMETRY
SECTION = A
Question: The shadow of a 30 m high tower on the ground at some time of the day is 
m long , then the angle of elevation of the sun at that time is :
(a) 30° (b) 90° (c) 45° (d) 60°
Solution: (d) 60° .
[ Here , 
and 
In 
we have , 
]
Question: The ratio of the height of a tower and the length of its shadow on the ground is 
, then the angle of elevation of the sun is : [CBSE2017]
(a) 60° (b) 30° (c) 70° (d) 90°
Solution: (a) 60°
[ Here , 
In 
we have , 
Therefore, the angle of elevation of the sun is 60° . ]
Question: The angle of the elevation of the top of a tower from a point on the ground ,which is 15m away from the foot of the tower, is 60° . The height of the tower is :[SEBA 2019]
(a) 15 m (b) 
m (c) 
m (d) 
m
Solution: (b) 
m
[ Here, 
m and
]
In we have , 
m
Therefore, the height of the tower is m ]
Question: The angle of elevation of the top of a tower from a point on the ground, which is 30 m away from the foot of the tower is 45° . The height of the tower is : [SEBA 2018]
(a) 30 m (b) 15 m (c) 10 m (d) 60 m
Solution: (a) 30 m
[ Here, 
m and 
In we have , 
Therefore, the height of the tower is 30 m . ]
Question: A pole casts a shadow of length 
m on the ground , when the sun’s elevation is 60° then the height of the pole is : [CBSE2015]
(a) 6 m (b) 8 m (c) 12 m (d) 10 m
Solution: (A) 6 m
[ Here , 
and 
m
In we have , 
Therefore, the height of the pole is 6 m . ]
Question: A ladder 15 m long makes an angle of 60° with the wall , then the height of the point where the ladder touches the wall is :
(a) 
m (b) 
m (c) 
m (d) 
m
Solution: (d) m
[ Here , 
and and 
m
In we have , 
]
Question: In figure, AB is a 6 m high pole and CD is a ladder inclined at an angle of 60° to the horizontal and reaches up to a point D of pole . If AD = 2.54 m , then the length of the ladder is ( Use 
) : [CBSE2016]
(a) 3 m (b) 2 m (c) 4 m (d) 8 m
Solution: (c) 4 m
[ Here , 
m , 
m , 
m and 
In 
we have , 
Therefore, the length of the ladder is 4 m . ]
Question: A ladder of length 
m reaches a window 15 m high , then the inclination of the ladder with the ground is :
(a) 30° (b) 45° (c) 60° (d) 90°
Solution: (b) 45° .
[ Here , 
m and
m
In , we have 
Therefore, the inclination of the ladder with the ground is 45° . ]
Question: The angle of depression of an object from the top of a tower of height 75 m is 30° .Then the distance of the object from the foot of the tower is : [SEBA 2017]
(a) 
m (b) 
m (c) 
m (d) 150 m
Solution: (c) m
[ Here, BC = height of the tower = 75 m , AB = the distance of the object from the foot of the tower .
In 
we have,
]
Question: If the angle of elevation of the sun is 45° , then the ratio between the tower and its shadow is : [ SEBA 2015 ]
(a) 
(b) 
(c) 
(d) 
Solution: (a) 1 : 1
[ Let, 
the height of the tower , 
the height of shadow and 
.
In we have ,
Therefore, the ratio between the tower and its shadow is 1 : 1 . ]
Question: When the sun is 30° above the horizontal the length of the shadow cast by 50 m building is :
(a) m (b) 
(c) m (d) 
Solution: (b)
[ Here , 
and 
m
In 
we have , 
Therefore, the length of the building is . ]
Question: A ladder , leaning against a wall, makes an angle of 60° with the horizontal . If the foot of the ladder is 2.5 m away from the wall, then the length of the ladder is :
(a) 2.5 m (b) 5.2 m (c) 
m (d) 5 m
Solution: (d) 5 m
[ Here , 
m and 
In we have , 
Therefore , the length of the ladder is 5 m . ]
Question: In figure, a tower AB is 20 m high and BC, its shadow on the ground , is 
m long , then the sun’s altitude is :
(a) 30° (b) 45° (c) 60° (d) 90°
Solution: (a) 30°
[ Here , 
m and 
m
In , we have 
]
Question: A tower stands vertically on the ground. From a point on the ground, which is 15 m away from the foot the tower, the angle of elevation of the top of the tower is found to be 30° .The height of the tower is: [SEBA 2020] (a) 
m (b) m (c) 15 m (d) 
Solution: (d)
[ Here , 
m and 
In we have , 
Therefore, the height of the tower is . ]
Question: The angle of elevation of the sun’s altitude when the height of the shadow of a vertical pole is equal to its height is :
(a) 90° (b) 30° (c) 45° (d) 60°
Solution: (c) 45°
[ Here , the height of the shadow and the height of the pole .
Given , 
In 
we have , 
Therefore, the angle of elevation is 45° . ]
Question: 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 : [CBSE 2014]
(a) 
(b) 
(c) 
(d) 
Solution: (a)
[ Here, BC = height of the tower = 150 m , AB = the distance of the car from the tower and .
In , we have 
]
Answers Following the Questions
Question: In figure, the angle of elevation of the top of a tower AC from a point B on the ground is 60° . If the height of the tower is 20 m , find the distance of the point from the foot of the tower . [ CBSE 2020 Basic]
Solution: Here, 
m , 
and the distance of the point from the foot of the tower .
In we have , 
Therefore, the distance of the point from the foot of the tower is 
m .
Question: In figure, the angle of elevation of the top of a tower from a point C on the ground, which is 30 m away from the foot of the tower is 30 . Find the height of the tower . [CBSE 2020 standard]
Solution: Here , m , 
and the height of tower .
In we have , 
Therefore, the height of the tower is 
m .
Question: If the height of a vertical pole is 
times the length of its shadow on the ground, then find the angle of elevation of the sun at that time . [CBSE 2014]
Solution: Given, 
In we have ,
Therefore, the angle of elevation of the sun at that time is 60° .
SECTION = C
Question: From a point on a bridge across a river, the angles of depression of tthe banks on opposite sides of the river are 30° and 45°, respectively. If the bridge is at a height of 3 m from the banks, find the width of the river.
Solution: Here, AB = the width of the river , DP = the height between the bridge and river = 3 m
In 
, we have 
In 
we have, 
Therefore, the width of the river is 
.
Question: 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 30° to 60° as he walks towards the building . Find the distance he walked towards the building .
Solution: In given figure :
Here, BC = the distance of the building and the boy , DE = the height of the building = 30 m
AE = BF = CG = 1.5 m and AD = 30 – 1.5 = 28.5 m
In 
we have, 
In we have, 
Therefore, the distance he walked towards the building is 
Question: From the top of a 7 m 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 .
Solution: In given figure :
Here, AB = CE = height of the building = 7 m , BC = AE = the distance between the tower and the building , 
and 
In 
we have , 
In we have , 
From 
and 
we get , 
Therefore, the height of the tower is 
.
Question: A tree breaks due to storm and the broken part bends so that the top of the tree touches the ground making an angle 30° with it . The distance between the foot of the tree to the point where the top touches the ground is 8 m . Find the height of the tree .
Solution: In given figure :
Here, AB = 8 m , AD = Height of the tree , BE = DE and the angle of elevation
In 
we have 
and 
From (i) and (ii) we get, 
From (i) we get , 
Therefore, the height of the tree is 
Question: The shadow of a tower standing on a level ground is found to be 40 m longer when the sun's altiitude is 30° than when it is 60° . Find the height of the tower .
Solution: Here, AB = the height of the tower , BC = the length of the shadow when the sun’s altitude is 60° and CD = 40 m .
In 
, we have
In 
, we have 
From (i) , we get 
Therefore, the height of the tower is 
.
Question: A statue, 1.6 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 .
Solution: In given figure :
Here, CD = 1.6 m , BC = the height of the pedestal , AB = the distance between the ground and the foot point of the pedestal and Angle of the elevation 
and 
.
In we have 
In we have, 
Therefore, the height of the pedestal is 
.
Question: The angle of elevation of the top of a building from the foot of the tower is 30° and the angle of elevation of the top of the tower from the foot of the building is 60° . If the tower is 50 m high ,find the height of the building
Solution: In given figure :
Here, AB = 50 m , CD = the height of the building , BC = the distance between the building and the tower .
The angle elevation are 
and 
In we have 
In 
we have 
From (i) and (ii) , we get 
Therefore, the height of the building is 
.
SECTION = D
Question: As observed from the top of a 75 m high lighthouse from the sea-level, the angles of depression of two ships are 30° and 45° . If one ship is exactly behind the other on the same side of the lighthouse, find the distance between the two ships.
Solution : In given figure:
Here, CD = the Height of lighthouse = 75 m and AB = The distance between the two ships.
The angle of elevation are 
and 
In we have , 
In 
we have, 
From i and ii we get , 
m
The distance between to the ship is 
m .
Question: From a point P on the ground the angle of elevation of the top of a 10 m tall building is 30°. A flag is hoisted at the top of the building and the angle of elevation of the top of the flagstaff from P is 45° .Find the length of the flagstaff and the distance of the building from the point P. [Use 
1.73 ]
Solution: In figure, AB = the height of the building = 10 m ; BD = the length of flagstaff .
Angle of elevation are 
In 
we have,
In 
we have
Therefore , the distance of the building from the point P is 17.32 m and the length of the flagstaff is 17.32 m .
Question: A straight highway leads to the foot of a tower . A man standing at 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 from this point.
Solution: In given figure :
We kow that, 
Let , 
m ; 
and 
In we have , 
In we have , 
i and ii we get , 
Therefore, the time taken by the car to reach the foot of the tower is 3 second .
Question: From a balloon vertically above a straight road , the angles of depression of two cars at an instant are found to be 45° and 60° . If the cars are 100 m apart , find the height of the balloon .
Solution: Let, 
(= PQ) be the height of the balloon .
In figure , AB = 100 m , PQ =
In 
, we have 
In 
, we have 
From i
and ii
, we get 
From , we get 
m
Therefore , the height of the balloon is 
m .
Question: 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 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° (see Fig. 9.12). Find the height of the tower and the width of the canal.
Solution: Here, AB = the height of the tower , CD = 20 m and BC = the width of the canal .
The angle of elevation are 
and 
In we have ,
In we have ,
From and 
we get, 
From we get, 
m
Therefore, the height of the tower is 
m and the width of the canal is 
m .
Question: Two poles of equal heights are standing opposite each other on 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° and 30° ,respectively . Find the height of the poles and the distances of the point from the poles .
Solution : In given figure :
Here, AE = CD = The height of the poles , AC = The distance between the two pole = 80 m
The angle of elevation, 
and 
In we have, 
In we have,
From , we get 
∴ 
Therefore, the height of the pole is m and the distance of the point from the poles are 
m and 
m.
Posted 5 years ago