Height and Distance Mathematic

Height and Distance Mathematic

Important information:- It is an important application of trigonometry which helps us to find the height of any object and distance of that object from any point which are not directly measurable. If the angle of elevation/ depression from a point is known.

Line of Sight:- A line of sight is the line drawn from the eye of an observer to the point, where the object is viewed by the observer.

Horizontal Line:- The line of sight which is parallel to ground level is known as horizontal line.

Angle of Elevation:- The angle of elevation of the point viewed is the angle formed by the line of sight with the horizontal, when the point being viewed is above the horizontal level.

Angle of Depression:- When the line of sight is below the horizontal level, the angle so formed by the line of sight with the horizontal is called the angle of depression .

Table of Contain:-

# The shadow of a tower is 15 m, when the Sun's elevation is 30°. What is the length of the shadow, when the Sun's elevation is 60° ?

#The tops of two poles of height 24 m and 36 m are connected by a wire. If the wire makes an angle of 60° with the horizontal, then the length of the wire is ?

#The angle of elevation of the top of an unfinished pillar at a point 150 m from its base is 30°. If the angle of elevation at the same point is to be 45°, then the pillar has to be raised to a height of how many meters?

1. What is the angle of elevation of the Sun when the shadow of a pole is √3 times the length of the pole ?

(a) 30°

(b) 45°

(c) 60°

(d) None of these

Answer :- (a) 30°

2. The shadow of a tower is 15 m, when the Sun's elevation is 30°. What is the length of the shadow, when the Sun's elevation is 60° ?

(a) 3 m

(b) 4 m

(c) 5 m

(d) 6 m

Answer :- (c) 5 m

3. What is the angle of elevation of the Sun, when the shadow of a pole of height x m is —m√3 ?

(a) 30°

(b) 45°

(c) 60°

(d) 75°

Answer :- (c) 60°

4. A vertical stick 12 m long casts a shadow 8 m long on the ground. At the same time, a tower casts a shadow of 40 m long on the ground. The height of the tower is [SSCCGL2012]

(a) 60 m

(b) 65 m

(c) 70 m

(d) 72 m

Answer :- (a) 60 m

5. The tops of two poles of height 24 m and 36 m are connected by a wire. If the wire makes an angle of 60° with the horizontal, then the length of the wire is [SSCCCL2012]

(a) 8√3 m

(b) 8 m

(c) 6√3 m

(d) 6 m

Answer :- (a) 8√3 m

6. The shadow of a tower standing on a level plane is found to be 50 m longer when the Sun's elevation is 30°. When it is 60°, then what is the height of the tower? [CDS 2014]

(a) 25m

(b) 25√3m

(c) 25/√3m

(d) 30m

Answer :- (c) 25/√3m

7. The angle of elevation of the tip of a tower from a point on the ground is 45°. Moving 21 m directly towards the base of the tower, the angle of elevation changes to 60°. What is the height of the tower, to the nearest meter?

(a) 48 m

(b) 49 m

(c) 50 m

(d) 51 m

Answer :- (c) 50 m

8. The angles of depression from the top of a light house of two boats are 45° and 30° towards the west. If the two boats are 5 m apart, then the height of the light house is [SSCFCI2012]

(a) (2.5√3 - 1) m

(b) 2.5(√3 - 1) m

(c) (2.5√3 + 1) m

(d) 2.5(√3 + 1) m

Answer :- (d) 2.5(√3 + 1) m

9. The angle of elevation of the top of an unfinished pillar at a point 150 m from its base is 30°. If the angle of elevation at the same point is to be 45°, then the pillar has to be raised to a height of how many meters?

(a) 59.4 m

(b) 61.4 m

(c) 62.4 m

(d) 63.4 m

Answer :- (d) 63.4 m

10. From the top of a cliff 200 m high, the angles of depression of the top and bottom of a tower are observed to be 30° and 45°, respectively. What is the height of the tower? [CDS 2012]

(a) 200 m

(b) 200{(√3 - 1)/ √3} m

(c) 200√3 m

(d) None of these

Answer :- (b) 200{(√3 - 1)/ √3} m

11. On walking 120 m towards a chimney in a horizontal line through its base the angle of elevation of tip of the chimney changes from 30° to 45°. The height of the chimney is ?

(a) 120 m

(b) 60(√3 - 1) m

(c) 60(√3 + 1) m

(d) None of these

Answer :- (c) 60(√3 + 1) m

12. A man standing at a point P is watching the top of elevation of 30°. The man walks some distance towards the tower and then his angle of elevation of the top of the tower is 60°. If the height of the tower is 30 m, then the distance he moves is ?

(a) 20 m

(b) 20√3 m

(c) 22 m

(d) 22√3 m

Answer :- (b) 20√3 m

13. The angle of elevation of the top of a tower from the bottom of a building is twice that from its top. What is the height of the building, if the height of the tower is 75 m and the angle of elevation of the top of the tower from the bottom of the building is 60°? [CDS 2011 ]

(a) 25 m

(b) 37.5 m

(c) 50 m

(d) 60 m

Answer :- (c) 50 m

14. The angles of elevation of the top of a tower from two points which are at distances of 10 m and 5 m from the base of the tower and in the same straight line with it are complementary. The height of the tower is ?

(a) 5 m

(b) 15 m

(c) √50 m

(d) √75 m

Answer :- (c) √50 m

15. The angles of elevation of e top of an inaccessible tower from two points on the same straight line from the base of the tower are 30° and 60°, respectively. If the points are separated at a distance of 100 m, then the height of the tower is close to ?

(a) 86.6 m

(b) 84.6 m

(c) 82.6 m

(d) 80.6 m

Answer :- (a) 86.6 m

16. Two poles of heights 6 m and 11 m stand on a plane ground. If the distance between their feet is 12 m, what is the distance between their tops?

(a) 13 m

(b) 17 m

(c) 18 m

(d)23 m

Answer :- (a) 13 m

17. If P = 30°, then what is the value of tan a? (a) 1/2 (b) 1/3 (c) 1/4 (d) None of these 18. If a = 30°, then what is the value of tan P?

(a) 1

(b) 1/2

(c) 1/3

(d) None of these

Answer :- (c) 1/3

19. If a = 30° and h = 30 m, then what is the distance between the base of the building and the base of the tower?

(a) 15 + 15√3 m

(b) 30 + 15√3 m

(c) 45 + 1 5√3 m

(d) None of these

Answer :- (c) 45 + 1 5√3 m

20. If P = 30° and if 6 is the angle of depression of the foot of the tower as seen from the top of the building, then what is tan 9 equal to?

(a)(3 -√3) / 3√3

(b)(2 - √3) / 3√3

(c) (3 + √3)/ 3√3

(d) None of these

Answer :- (a)(3 -√3) / 3√3

21. If the angle of elevation of a tower from two distant points a and b (a > b) from its foot and in the same straight line and on the same side of it are 30° and 60°, then the height of the tower is ?

(a) √(ab)

(b) √(a / b)

(c) √(a + b)

(d) √(a - b)

Answer :- (a) √(ab)

22. The angle of elevation of the top of a tower 30 m high from the foot of another tower in the same plane is 60° and the angle of elevation of the top of the second tower from the foot of the first tower is 30°. The distance between the two towers is n times the height of the shorter tower. What is n equal to?

(a) √2

(b) √3

(c) 1/2

(d) 1/3

Answer :- (b) √3

23. At the foot of a mountain, the elevation of its summit is 45°. After ascending 2 km towards the mountain upon an incline of 30°, the elevation changes to 60°. The height of the mountain is

(a) (√3 + 2) km

(b) (√3 - 2) km

(c) (√3 +1) km

(d) (√3 - 1) km

Answer :- (c) (√3 +1) km

24. A man standing in one corner of a square football field observes that the angle subtended by a pole in the corner just diagonally opposite to this corner is 60°. When he retires 80 m from the corner, along the same straight line, he finds the angle to be 30°. The length of the field is ?

(a) 20m

(b) 40m

(c) 20√2m

(d) 40√2m

Answer :- (b) 40m

25. A spherical balloon of radius r subtends angle 60° at the eye of an observer. If the angle of elevation of its center is 60° and h is the height of the centre of the balloon, then which one of the following is correct? [CDS 2013]

(a) h = √3r

(b) h = √2r

(c) h = 4r

(d)h = 2r

Answer :- (a) h = √3r