Chapter 9: Some Applications of Trigonometry — Class 10 Mathematics NCERT Solutions | CBSE
Chapter 9: Some Applications of Trigonometry – NCERT Class 10 Mathematics Solutions | CBSE
Chapter 9: Some Applications of Trigonometry
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Chapter 9 : Some Applications of Trigonometry |
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Exercise 9.1 complete solution |
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1. (i) The line of sight is the line drawn from the eye of an observer to the point in the object viewed by the observer. In figure , |
EXERCISE 9.1
1. A circus artist is climbing a 20m long rope, which is tightly stretched and tied from the rope of a vertical pole to the ground . Find the height of the rope , if the angle made by rope with the ground level is 30° (See Fig. 9.11) .
Solution: Here, the length of the rope 
, AB = the height of the pole and the angle of elevation ∠ACB=30° .
In 
we have ,
Therefore, the height of the pole is 10 m .
2. 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 
3. Acontractor plans to install two slides for the children to play in a park . For the children below the age of 5 years , she prefers to have a slide whose top is at a height of 1.5 m , and is inclined at an angle of 30° to the ground, whereas for elder children, she wants to have a steep slide at a height of 3 m , and inclined at an angle of 60° to the ground . What should be the lenght of the slide in each case ?
Solution: In given figure :
For the children below the age of 5 years :
Here, AE = 1.5 m = height of the slide, CE = the length of the slide and the angle of elevation 
In 
we have
For elder children : Here, AB = 3 m = the height of the slide , the angle of elevation 
and BD = the length of the slide .
In we have
Hence, the length of the two slides are 3 m and 
.
4. 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 30° . Find the height of the tower .
Solution: In given figure:
Given, 
m .
In 
, we have 
Thus , the height of the tower is 
.
5. A kite is flying at a height of 60 m above the ground . The string attached to the kite is temporarily tied to a point on the ground . The inclination of the string with the ground is 60° . Find the length of the string , assuming that there is no slack in the string .
Solution: In given figure :
Here, BC = the height between the ground and kite = 60 m and the angle of elevation 
.
AC = The length of the string .
In , We have 
Therefore, The length of the string is 
6. 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 
7. 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 45° and 60° respectively . Find the height of the tower .
Solution: In given figure :
Here, BC = height of the building = 20 m , CD = the height of the tower .
In we have , 
In 
we have , 
[From ()]
Therefore, the height of the tower is 
8. 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 
.
9. 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 
.
10. 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.
11. 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 .
12. 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 
.
13. 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 .
14. A 1.2 m tall girl spots a balloon moving with the wind in a horizontal line at a height of 88.2 m from the ground . The angle of elevation of the balloon from the eyes of the girl at any instant is 60° .After some time, the angle of elevation reduces to 30° (see Fig. 9.13) . Find the distance travelled by the balloon during the interval .
Solution: Here, AF = BG = CH = 1.2 m , GD = HE = 88.2 m , BC = distance between the position of the two balloon and BD = CE = 88.2 – 1.2 = 87 m
In We have 
In 
we have 
Therefore, the distance travelled by the balloon during the interval is 
.
15. 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 .
16. 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 : In given figure :
Let DC = the height of the tower , AC = 9 m , BC = 4 m and Angle of elevations are 
and 
.
In , we have 
In , we have 
Multiplying and , we get 
m [ only positive value]
Therefore, the height of the tower is 6 m . Proved.
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