Height and Distance Aptitude

Height and Distance problems are an integral part of the trigonometry section in quantitative aptitude tests. These problems typically involve the application of basic trigonometric ratios (sine, cosine, and tangent) to find heights and distances in various scenarios. They require a clear understanding of angles of elevation and depression and how these angles are used in conjunction with trigonometric identities to solve real-world problems.

Height and Distance – Formulas

1. From the top of a 7-meter high building, the angle of elevation of the top of a tower is 60 degrees. If the tower and the building are 8 meters apart, what is the height of the tower?

Let the height of the tower be H meters.
Angle of elevation = 60 degrees.
Distance between tower and building = 8 meters.
Height of building = 7 meters.
Using tangent, tan(60) = (H - 7) / 8.
√3 = (H - 7) / 8.
H - 7 = 8√3.
H = 8√3 + 7 ≈ 14 meters.

2. A ladder is placed against a wall such that its top reaches the top of the wall at an angle of 45 degrees. If the height of the wall is 10 meters, what is the length of the ladder?

Height of the wall = 10 meters.
Angle of elevation = 45 degrees.
Using sine, sin(45) = Opposite / Hypotenuse = 10 / Length of ladder.
1/√2 = 10 / Length of ladder.
Length of ladder = 10√2 meters.

3. From a point on the ground, the angle of elevation of the top of a building is 30 degrees. If the building is 20 meters high, how far is the point from the base of the building?

Height of the building = 20 meters.
Angle of elevation = 30 degrees.
Using tangent, tan(30) = Opposite / Adjacent = 20 / Distance.
1/√3 = 20 / Distance.
Distance = 20√3 meters.

4. The angle of elevation of the top of a tree from a point on the ground is 45 degrees. If the point is 12 meters away from the base of the tree, what is the height of the tree?

Distance from the tree = 12 meters.
Angle of elevation = 45 degrees.
Using tangent, tan(45) = Opposite / Adjacent = Height / 12.
1 = Height / 12.
Height = 12 meters.

5. A pole casts a shadow of 8 meters when the angle of elevation of the sun is 60 degrees. What is the height of the pole?

Length of shadow = 8 meters.
Angle of elevation of the sun = 60 degrees.
Using tangent, tan(60) = Opposite / Adjacent = Height / 8.
√3 = Height / 8.
Height = 8√3 meters.

6. From the top of a tower, the angle of depression to a point on the ground is 30 degrees. If the tower is 50 meters high, how far is the point from the base of the tower?

Height of the tower = 50 meters.
Angle of depression = 30 degrees.
Using tangent, tan(30) = Opposite / Adjacent = 50 / Distance.
1/√3 = 50 / Distance.
Distance = 50√3 meters.

7. A person standing 30 meters away from a flagpole observes the angle of elevation of the top of the pole as 60 degrees. What is the height of the flagpole?

Distance from the flagpole = 30 meters.
Angle of elevation = 60 degrees.
Using tangent, tan(60) = Opposite / Adjacent = Height / 30.
√3 = Height / 30.
Height = 30√3 meters.

8. The angle of elevation of the top of a hill at the foot of the tower is 30 degrees and the angle of elevation of the top of the tower from the foot of the hill is 60 degrees. If the tower is 20 meters high, what is the height of the hill?

Height of the tower = 20 meters.
Angle of elevation from hill to tower = 60 degrees.
Let the height of the hill be H meters.
Using tangent, tan(60) = Opposite / Adjacent = 20 / (H - 20).
√3 = 20 / (H - 20).
Solving for H, H = 10√3 meters.

9. A balloon is connected to a 60-meter long string which makes an angle of 60 degrees with the ground. What is the height of the balloon from the ground?

Length of string = 60 meters.
Angle with the ground = 60 degrees.
Using sine, sin(60) = Opposite / Hypotenuse = Height / 60.
√3/2 = Height / 60.
Height = 30√3 meters.

10. From the top of a cliff 150 meters high, the angles of depression of the top and bottom of a tower are observed to be 30 degrees and 60 degrees respectively. What is the height of the tower?

Height of the cliff = 150 meters.
Angle of depression to the top of the tower = 30 degrees.
Angle of depression to the bottom of the tower = 60 degrees.
Let the height of the tower be H meters.
Using tangent for top of the tower, tan(30) = (150 - H) / Base.
1/√3 = (150 - H) / Base.
Using tangent for bottom of the tower, tan(60) = 150 / Base.
√3 = 150 / Base.
Solving the two equations, H = 75 meters.

11. The angle of elevation of a cloud from a point 60 meters above a lake is 30 degrees and the angle of depression of its reflection in the lake is 60 degrees. What is the height of the cloud above the lake?

Let the height of the cloud above the lake be H meters.
Angle of elevation = 30 degrees, angle of depression = 60 degrees.
Using tangent for elevation, tan(30) = H / Base.
1/√3 = H / Base.
Using tangent for depression, tan(60) = (H + 60) / Base.
√3 = (H + 60) / Base.
Solving the two equations, H = 120 meters.

12. A person observes the top of a tower at an angle of elevation of 45 degrees. After walking 40 meters towards the tower, the angle of elevation is 60 degrees. What is the height of the tower?

Let the height of the tower be H meters.
Initial angle of elevation = 45 degrees, distance = D meters.
Final angle of elevation = 60 degrees, distance = D - 40 meters.
Using tangent for initial position, tan(45) = H / D.
1 = H / D.
Using tangent for final position, tan(60) = H / (D - 40).
√3 = H / (D - 40).
Solving the two equations, H = 40 meters.

13. The angle of elevation of the top of a building from the foot of the tower is 30 degrees and from the top of the tower, which is 40 meters high, is 60 degrees. What is the height of the building?

Height of the tower = 40 meters.
Angle of elevation from foot = 30 degrees, angle from top = 60 degrees.
Let the height of the building be H meters.
Using tangent for angle from foot, tan(30) = H / Base.
1/√3 = H / Base.
Using tangent for angle from top, tan(60) = (H - 40) / Base.
√3 = (H - 40) / Base.
Solving the two equations, H = 60 meters.

14. A man standing at a point P is watching the top of a tower, which makes an angle of elevation of 30 degrees with the man's eye. The man walks some distance towards the tower to watch its top and the angle of the elevation becomes 60 degrees. If the height of the tower is 20√3 meters, what is the distance between the base of the tower and point P?

Height of the tower = 20√3 meters.
Initial angle of elevation = 30 degrees, final angle = 60 degrees.
Using tangent for initial angle, tan(30) = 20√3 / Distance.
1/√3 = 20√3 / Distance.
Distance = 20 meters.

15. The shadow of a tower standing on a level plane is found to be 40 meters longer when the sun's altitude is 30 degrees than when it is 60 degrees. What is the height of the tower?

Let the height of the tower be H meters.
Using tangent for 30 degrees, tan(30) = H / (Shadow + 40).
1/√3 = H / (Shadow + 40).
Using tangent for 60 degrees, tan(60) = H / Shadow.
√3 = H / Shadow.
Solving the two equations, H = 40 meters.

16. 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 of 30 degrees with the ground. If the top of the tree touches the ground 10 meters away from the base of the tree, what was the original height of the tree?

Distance from the base = 10 meters.
Angle formed = 30 degrees.
Original height = Broken part's length.
Using tangent, tan(30) = Opposite / Adjacent = Height / 10.
1/√3 = Height / 10.
Height = 10√3 meters.
Using Pythagoras theorem, Original height = √(Height^2 + Base^2) = √(300 + 100) = √400 = 20 meters.

17. The angle of elevation of a ladder leaning against a wall is 60 degrees and the foot of the ladder is 4.6 meters away from the wall. What is the length of the ladder?

Distance from the wall = 4.6 meters.
Angle of elevation = 60 degrees.
Using cosine, cos(60) = Adjacent / Hypotenuse = 4.6 / Length of ladder.
1/2 = 4.6 / Length of ladder.
Length of ladder = 4.6 * 2 = 9.2 meters.

18. A boy is standing at a distance of 50 meters from a building. The angle of elevation from his eyes to the top of the building increases from 30 degrees to 45 degrees as he walks towards the building for 20 meters. Find the height of the building.

Initial distance from building = 50 meters.
Distance walked towards building = 20 meters.
Final distance from building = 50 - 20 = 30 meters.
Initial angle of elevation = 30 degrees, final angle = 45 degrees.
Using tangent for final angle, tan(45) = Height / 30.
1 = Height / 30.
Height = 30 meters.

19. From a point on the ground, the angle of elevation of the bottom and the top of a transmission tower fixed at the top of a 20-meter high building are 45 degrees and 60 degrees respectively. What is the height of the transmission tower?

Height of the building = 20 meters.
Angle of elevation to the bottom of the tower = 45 degrees.
Angle of elevation to the top of the tower = 60 degrees.
Using tangent for the bottom of the tower, tan(45) = 20 / Base.
1 = 20 / Base.
Base = 20 meters.
Using tangent for the top of the tower, tan(60) = (Height of tower + 20) / 20.
√3 = (Height of tower + 20) / 20.
Solving for Height of tower, Height = 20 meters.

20. The shadow of a pole is 2√3 meters long when the angle of elevation of the sun is 60 degrees. What is the height of the pole?

Length of shadow = 2√3 meters.
Angle of elevation of the sun = 60 degrees.
Using tangent, tan(60) = Opposite / Adjacent = Height / 2√3.
√3 = Height / 2√3.
Height = 2√3 * √3 = 6 meters.

21. A person standing at a distance of 80 meters from a tower observes the angle of elevation as 30 degrees. What is the height of the tower?

Distance from the tower = 80 meters.
Angle of elevation = 30 degrees.
Using tangent, tan(30) = Opposite / Adjacent = Height / 80.
1/√3 = Height / 80.
Height = 80 / √3 = 20√3 meters.

22. The angles of elevation of the top of a tower from two points at a distance of a and b meters from the base of the tower and in the same straight line with it are complementary. What is the height of the tower?

Let the angles be θ and 90 - θ.
Using tangent for first angle, tan(θ) = Height / a.
Using tangent for second angle, tan(90 - θ) = Height / b.
Since tan(90 - θ) = cot(θ), we have 1/tan(θ) = Height / b.
Therefore, a / Height = b / Height.
Height = ab / (a - b).

23. A kite is flying at a height of 75 meters, attached to a string inclined at an angle of 60 degrees to the horizontal. What is the length of the string?

Height of kite = 75 meters.
Angle of inclination = 60 degrees.
Using sine, sin(60) = Opposite / Hypotenuse = 75 / Length of string.
√3/2 = 75 / Length of string.
Length of string = 75 / (√3/2) = 75√3 meters.

24. The angle of elevation of a cloud from a point 100 meters above a lake is 45 degrees. What is the height of the cloud above the lake?

Height above the lake = 100 meters.
Angle of elevation = 45 degrees.
Using tangent, tan(45) = Height / 100.
1 = Height / 100.
Height = 100 meters above the point.
Total height above the lake = 100 + 100 = 200 meters.

25. A tower casts a shadow of 50 meters when the angle of elevation of the sun is 45 degrees. What is the height of the tower?

Length of shadow = 50 meters.
Angle of elevation of the sun = 45 degrees.
Using tangent, tan(45) = Opposite / Adjacent = Height / 50.
1 = Height / 50.
Height = 50 meters.