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SEBA Class 10 Mathematics Chapter 9 Some Application Trigonometry
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Some Application Trigonometry
Chapter – 9
| Exercise 9.1 |
1. A circus artist is climbing a 20 m long rope, which is tightly stretched and tied from the top of a vertical pole to the ground. Find the height of the pole, if the angle made by the rope with the ground level is 30°.
Ans: It can be observed from the figure that AB is the pole.
In ∆ABC,
Therefore, the height of the pole is 10 m.
2. A tree breaks due to a 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. A contractor 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 1.5m., and is inclined at an angle of 30° to the ground, whereas for elder children, she wants to have a step side at a height of 3m. and inclined at an angle of 60° to the ground. What should be the length 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 30m away from the foot of the tower is 30°. Find the height of the tower.
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.
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.
Ans:
Let the boy was standing at point S initially. He walked towards the building and reached at point T. It can be observed that
PR = PQ − RQ
= (30 − 1.5) m = 28.5 m
In ∆PAR, 57/2 m
ST = AB
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.
Ans:
Let BC be the building, AB be the transmission tower, and D be the point on the ground from where the elevation angles are to be measured.
In ∆BCD,
Therefore, the height of the transmission tower is 20(√3 − 1) m.
8. A 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 60° and from the same point the angle of elevation of the top of the pedestal is 45°. Find the height of the pedestal.
Ans:
Let AB be the statue, BC be the pedestal, and D be the point on the ground from where the elevation angles are to be measured.
In ∆BCD,
Therefore, the height of the pedestal is 0.8(√3 + 1) m.
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.
Ans:
Let AB be the building and CD be the tower. In ∆CDB,
Therefore, the height of the building is 16 2/3m.
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 30º, respectively. Find the height of poles and the distance of the point from the poles.
Ans:
Let AB and CD be the poles and O is the point from where the elevation angles are Measured.
In ∆ABO,
CD = AB
DO = BD − BO = (80 − 20) m = 60 m
Therefore, the height of poles is 20√3 m and the point is 20 m and 60 m far from these Poles.
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.
Ans:
In ∆ABC,
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.
Let AB be a building and CD be a cable tower.
In ∆ABD,
Therefore, the height of the cable tower is 7(√3 + 1) m.
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.
Ans:
Let AB be the lighthouse and the two ships be at point C and D respectively.
In ∆ABC,
Therefore, the distance between the two ships is 75(√3 − 1) 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 sometime, the angle of elevation reduces to 30°. Find the distance travelled by the balloon during the interval.
Ans:
Let the initial position A of the balloon change to B after some time and CD be the girl. In ∆ACE,
15. A straight highway leads to the foot of a tower. A man standing at the top of the tower observes a car an angle of depression of 30°, which is approaching the foot of the tower with 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.
Ans:
Let AB be the tower.
Initial position of the car is C, which changes to D after six seconds.
In ∆ADB,
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.
Ans:
Let AQ be the tower and R, S are the points 4m, 9m away from the base of the tower Respectively.
The angles are complementary. Therefore, if one angle is θ, the other will be 90 − θ.
In ∆AQR,
However, height cannot be negative.
Therefore, the height of the tower is 6 m.
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