Long Answer Questions: Some Applications of Trigonometry

# Class 10 Maths Chapter 9 Question Answers - Some Applications of Trigonometry

Q1. A boy whose eye level is 1.3 m from the ground, spots a balloon moving with the wind in a horizontal level at some height from the ground. The angle of elevation of the balloon from the eyes of the boy at any instant is 60°. After 2 seconds, the angle of elevation reduces to 30°. If the speed of the wind at that moment is 29√3 m/s, then find the height of the balloon from ground.

Sol. Let E and D be the two positions of the balloon.

Let AB be the position of the boy.
∴ AB = 1.3 m
⇒ HF = CG = 1.3 m
Also speed of the wind = 29√3 m/s
Distance covered by the balloon in 2 seconds
= ED = HG = 2 × 29√3 m
= 58√3 m
∴ AG = AH + HG
= AH + 58√3 m ...(1)
Now, in right Δ AEH, we have

Thus, the height of the balloon = 88.3 m.

Q2. From a point on the ground, the angles of elevation of the bottom and the top of a tower fixed at the top of a 20 m high building are 45° and 60° respectively. Find the height of the tower.

Sol. Let AB = 20 m be the building, P be the point on ground, and AC = xm be the tower.

Thus, the required height of the tower is 14.64 m

Q3. A statue, 1.5 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 45° and from the same point the angle of elevation of the top of the pedestal is 30°.
Find the height of the pedestal from the ground.

Sol. Let AB be the pedestal and AB = h
Let C be the point on the ground such that
BC = x metres.
In right Δ ACB, we have:

Q4. The angles of depression of the top and battom of an 8 m tall building from the top of a multi-storeyed building are 30° and 45°, respectively. Find the height of the multistoreyed building and the distance between the two buildings.

Sol. Let the multistoreyed building be AB.
∴ AB = q metres
⇒ AD =(q − 8) m [∵ BD = 8 m]
Let EC be the small building.
Now, in right Δ ABC, we have:

∴ Distance between the two buildings = 18.928 m
Height of the multi-storeyed building = 18.928 m.

Q5. From the top of a building 60 m high, the angles of depression of the top and bottom of a vertical lamp post are observed to be 30° and 60° respectively. Find:

(i) The horizontal distance between the building and the lamp post.
(ii) The height of the lamp post.

[Take √3 = 1.732]

Sol. In the figure, let CE be the building and AB be the lamp post
∴ CE = 60 m

= 60 m − 20
m = 40 m.
Also, the distances between the lamp post and the building
= 20√3 m = 20 × 1.732 m    [∵ 3 = 1.732]

= 34.64 m

Q6. The angle of elevation of a cloud from a point h meters above the surface of a lake is q and the angle of depression of its reflection in the lake is φ. Prove that the height of the clouds above the lake is

Sol. Let P be the cloud and Q be its reflection in the lake. As shown in the figure, let A be the point of observation such that AB = h
Let the height of the cloud above the lake = x
Let AL = d
From rt ΔPLA,  tan θ =

Q7. From a point 100 m above a lake, the angle of elevation of a stationary helicopter is 30° and the angle of depression of reflection of the helicopter in the lake is 60°. Find the height of the helicopter.

Sol. In the figure, A is the stationary helicopter and F is its reflection in the lake.

In right Δ AED, we have:

Thus, the height of the stationary helicopter = 200 m.

Q8. The angle of elevation of an aeroplane from a point on the ground is 60°. After a flight of 15 seconds, the angle of elevation changes to 30°. If the aeroplane is flying at a constant height of 1500√3 m, find the speed of the aeroplane.

Sol. In the figure, let E and C be the two locations of the aeroplane.
Height BC = ED
= 1500√3 m

In right Δ ABC, we have:

∴ Speed of the aeroplane
= 200 m/s.

Q9. A spherical balloon of radius r subtends an angle q at the eye of the observer. If the angle of elevation of its centre is φ, find the heights of centre of the balloon.

Sol. In the figure, let O be the centre of the balloon, and A be the eye of the observer. r be the radius.

∴ OP = r and PAQ = θ
Also, ∠OAB = φ
Let the height of the centre of the balloon be ‘h’ ⇒ OB = h.
InΔOAP, ∠OPA = 90°

Q10. As observered from the top of a light house, 100 m high above sea level, the angle of depression of a ship sailing directly towards it, changes from 30° to 60°. Determine the distances travelled by the ship during the period of observation. [Use√3 = 1.732]

Sol. Let A represents the position of the observer such that AB = 100 m
∴ In right Δ ABC, we have

Q11. From the top of a tower, 100 m high, a man observes two cars on the opposite sides of the tower with angles of depression 30° and 45° respectively. Find the distances between the cars. [use√3 = 1.73]

Sol. Let 'P' be the position of the observer.
A and B represent the cars. Height of the tower PQ = 100 m In rt Δ PQA,

Thus, distance between the cars A and B :

Q12. The angles of elevation and depression of the top and the bottom of a tower from the top of a building, 60m high, are 30° and 60° respectively. Find the difference between the heights of the building and the tower and the distance between them.

Sol. Let AB is building = 60 m and DC is the tower

Substituting the value of x from (2) in (1), we have :

⇒ Difference between the heights of building and tower = 20 m
Distance between the tower and building

The document Class 10 Maths Chapter 9 Question Answers - Some Applications of Trigonometry is a part of the Class 10 Course Mathematics (Maths) Class 10.
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## Mathematics (Maths) Class 10

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## FAQs on Class 10 Maths Chapter 9 Question Answers - Some Applications of Trigonometry

 1. What are some real-life applications of trigonometry?
Ans. Trigonometry has various real-life applications, such as finding the height of a building using angle of elevation, navigation using latitude and longitude, calculating distances and angles in surveying, determining the distance between two objects using trigonometric ratios, and analyzing the motion of waves using trigonometric functions.
 2. How can trigonometry be used in engineering and architecture?
Ans. Trigonometry plays a crucial role in engineering and architecture. It is used in designing structures, calculating the dimensions and angles of various components, determining the stability and load-bearing capacity of structures, analyzing forces acting on objects, and optimizing the design of mechanical systems.
 3. Can trigonometry be used in astronomy and space exploration?
Ans. Yes, trigonometry is extensively used in astronomy and space exploration. It helps in calculating the distance between celestial bodies, determining the size and shape of planets and stars, analyzing the motion of celestial objects, predicting eclipses, and understanding the behavior of light in space.
 4. How is trigonometry applied in physics and engineering?
Ans. Trigonometry is widely applied in physics and engineering. It helps in analyzing the motion of objects, calculating forces and velocities, understanding the behavior of waves and vibrations, determining the angles and distances in projectile motion, and solving problems related to rotational motion, electricity, and magnetism.
 5. What are some applications of trigonometry in everyday life?
Ans. Trigonometry has numerous applications in everyday life. It is used in navigation and GPS systems, calculating distances and angles in sports like golf and cricket, determining the height and range of objects, analyzing sound and light waves, designing roller coasters and amusement park rides, and even in art and design to create visual effects and perspective.

## Mathematics (Maths) Class 10

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