Chapter 9. Some applications of Trignometry
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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.
(ii) The angle of elevation of an object viewed, is the angle formed by the line of sight with the horizontal when it is above the horizontal level, i.e., the case when we raise our head to look at the object.
In figure ,  Angle of elevetion .
(iii) The angle of depression of an object viewed, is the angle formed by the line of sight with the horizontal when it is below the horizontal level, i.e., the case when we lower our head to look at the object.
2. The height or length of an object or the distance between two distant objects can be determined with the help of trigonometric ratios.
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Class 10 Maths Chapter 9. Some applications of Trignometry Exercise 9.1 Solutions
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 
, the height of the pole 
and the angle of elevation 
.
In 
we have ,
Therefore, the height of the rope 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: Given, AB = 30 m .
In given figure ,
In ∆ABC , we have,
Therefore, 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 (i)]
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 .
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° . Find the height of the tower and the width of the canal.
Solution: In given figure :
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 ii
we get, 
m
Therefore, the height of the tower is 
m and the width of the canal 
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 , 
m
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 
and 
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: In given figure,
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 ,
let the speed of the car is x m/s .
Here 
m ; 
and 
[ 
]
In we have ,
In
We have ,
and 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 .
