9. SOME APPLICATION OF TRIGONOMETRY (NCERT)
NCERT
9. Some applications of Trignometry
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 ,
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Angle of elevetion .
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) .
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 .
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 ?
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, 
m .
In
ABC , we have , 
m
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:
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:
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:
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:
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:
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 : Here, AB = CD = The height of the poles ;
BD= The distance between the two pole 
m
The angle of elevation , 
and 
In 
we have, 
In 
we have, 
m
From 
, we get 
m
m
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: Here, 
The height of the tower ,
20 m and 
The width of the canal .
The angle of elevation are 
and 
In 
we have, 
In
we have , 
From and 
we get, 
m
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: Here, 
height of the building 
m ;
the distance between tower and building ;
and 
In 
we have , 
In 
we have , 
m
m 
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 : Here, High of lighthouse 
m and 
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 .
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: Let 
m ; 
and 
[ Speed 
]
In
we have , 
In we have , 
and we get , 
second
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 : let AB = the height of the tower ,
BQ = 9 m , BP = 4 m
and Angle of elevations are 
AQB = 
and APB = 90° –
InAQB , we have
InAPB , we have
Multiplying and , we get
m [ only positive value]
Therefore, the height of the tower is 6 m .
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