Heights & Distances – Solved Problems with Step-by-Step Solutions

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Heights & Distances – Solved Problems

Heights and distances is a practical application of trigonometry. It involves calculating heights or distances using angles of elevation and depression. These problems are commonly asked in school exams and competitive examinations.

Problem 1: Height of a Tower

The angle of elevation of the top of a tower from a point on the ground is 45°. If the point is 10 meters away from the foot of the tower, find the height of the tower.

Solution:

Let the height of the tower be h meters.

tan 45° = h / 10

1 = h / 10 ⇒ h = 10 m

Answer: Height of the tower = 10 m

Problem 2: Angle of Elevation

The height of a building is 20 meters. Find the angle of elevation of the top of the building from a point on the ground 20 meters away from the foot of the building.

Solution:

tan θ = 20 / 20 = 1

θ = 45°

Answer: Angle of elevation = 45°

Problem 3: Finding Distance

The angle of elevation of the top of a tower is 30°. If the height of the tower is 10√3 meters, find the distance of the point of observation from the tower.

Solution:

tan 30° = Height / Distance

1/√3 = 10√3 / d

d = 30 m

Answer: Distance = 30 m

Problem 4: Angle of Depression

The angle of depression of a car from the top of a tower is 30°. If the height of the tower is 20 meters, find the distance of the car from the foot of the tower.

Solution:

Angle of depression equals angle of elevation.

tan 30° = 20 / d

d = 20√3 m

Answer: Distance = 20√3 m

Problem 5: Two Observations

The angles of elevation of the top of a tower from two points on the same side of the tower and in the same straight line are 30° and 60° respectively. If the distance between the two points is 10 meters, find the height of the tower.

Solution:

Let height of the tower = h meters Distance of nearer point = x

tan 60° = h / x ⇒ h = x√3

tan 30° = h / (x + 10) ⇒ h = (x + 10)/√3

Equating both:

x√3 = (x + 10)/√3

3x = x + 10 ⇒ x = 5

Height = 5√3 meters

Answer: Height of the tower = 5√3 m

Problem 6: Flagstaff on a Tower

The angle of elevation of the top of a flagstaff standing on a tower is 45°, while the angle of elevation of the top of the tower is 30°. If the height of the tower is 10 meters, find the height of the flagstaff.

Solution:

Let height of flagstaff = h meters Let distance of observation point = d

tan 30° = 10 / d ⇒ d = 10√3

tan 45° = (10 + h) / d

1 = (10 + h) / (10√3)

10 + h = 10√3 ⇒ h = 10(√3 − 1)

Answer: Height of flagstaff = 10(√3 − 1) m

Problem 7: Angle of Depression from a Cliff

From the top of a cliff 50 meters high, the angles of depression of two points on the horizontal ground in the same straight line and on the same side of the cliff are 30° and 60° respectively. Find the distance between the two points.

Solution:

Distance₁ = 50 / tan 30° = 50√3

Distance₂ = 50 / tan 60° = 50/√3

Distance between points:

50√3 − 50/√3 = 100/√3 m

Answer: Distance = 100/√3 m

Conclusion: Heights and distances problems strengthen the practical application of trigonometry. Understanding angles of elevation and depression and choosing the correct trigonometric ratio are key to solving these problems efficiently.

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