Some Applications of Trigonometry — Class 10 Maths practice questions

Question 1

A lighthouse is 12 m tall. From the top of the lighthouse, the angle of elevation of the top of a nearby cliff is 60° and the angle of depression of the foot of the cliff is 30°. What is the height of the cliff?

Accepted Answer

Option (A) is correct.

**Explanation:** Let the horizontal distance between the lighthouse and the cliff be d m. Using tan 30° = 12/d ⟹ d = 12√3 m. Let the height of the cliff above the top of the lighthouse be h m; using tan 60° = h/d ⟹ h = 12√3 × √3 = 36 m. ∴ Total height of cliff = 12 + 36 = **48 m**.

Question 2

A bird is perched on a lamp post between two friends Aryan and Meera, who are standing on the same horizontal ground on opposite sides of the lamp post. The angle of elevation of the bird from Aryan's position is 60° and the angle of elevation of the bird from Meera's position is 45°. If the height of the bird above the ground is 12 m, what is the distance between Aryan and Meera? (Use √3 = 1.73)

Accepted Answer

Option (B) is correct.

Explanation: Using tan θ = height / horizontal distance, Aryan's distance from the post: tan 60° = 12/dA ⟹ √3 = 12/dA ⟹ dA = 12/√3 = 4√3 m. Meera's distance from the post: tan 45° = 12/dM ⟹ 1 = 12/dM ⟹ dM = 12 m. Since Aryan and Meera stand on opposite sides, total distance = 4√3 + 12 m.

Question 3

A drone is hovering at a height of 8 m above the ground between two observation posts P and Q on the same horizontal ground. The angles of depression of posts P and Q from the drone are 45° and 30° respectively. What is the distance PQ (the distance between the two observation posts)?

Accepted Answer

Option (B) is correct.

**Explanation:** Let the drone hover at point D, height 8 m above the ground, with the foot of the perpendicular at O. Using tan θ = opposite/adjacent: for post P, angle of depression = 45° ⟹ OP = 8/tan 45° = 8/1 = 8 m; for post Q, angle of depression = 30° ⟹ OQ = 8/tan 30° = 8/(1/√3) = 8√3 m. Since the drone lies between P and Q, PQ = OP + OQ = 8 + 8√3 = 8(1 + √3) m.

Question 4

From a point on the ground, the angles of elevation of the bottom and the top of a flagpole fixed at the top of a 30 m high tower are 30° and 45° respectively. What is the height of the flagpole? (Use √3 = 1.73)

Accepted Answer

Option (B) is correct.

**Explanation:** Let P be the point on the ground, B the base of the tower, T the top of the tower (= bottom of flagpole), and F the top of the flagpole. Let the horizontal distance PB = d and the height of the flagpole = h, so total height BF = (30 + h) m.

Using tan 30° = 30/d ⟹ d = 30√3 m.

Using tan 45° = (30 + h)/d ⟹ 30 + h = d = 30√3 ⟹ h = 30√3 − 30 = 30(√3 − 1) = 30(1.73 − 1) = 30 × 0.73 = **21.9 m**.

Question 5

Two vertical poles are standing on the same horizontal ground. The angle of elevation of the top of the taller pole from the foot of the shorter pole is 60°, and the angle of elevation of the top of the shorter pole from the foot of the taller pole is 30°. If the height of the taller pole is 27 m, what is the height of the shorter pole?

Accepted Answer

Option (B) is correct.

**Explanation:** Let H = 27 m (taller pole), h = shorter pole height, d = distance between poles. tan 60° = H/d ⟹ √3 = 27/d ⟹ d = 27/√3 = 9√3 m. tan 30° = h/d ⟹ 1/√3 = h/(9√3) ⟹ h = 9√3/√3 = 9 m. ∴ Height of the shorter pole = **9 m**.

Question 6

A vertical pole of height 15 m stands on a horizontal plane. A person observes the top of the pole from a point on the ground. The angle of elevation of the top of the pole from the point of observation is 30°. Find the distance of the person from the base of the pole. (Use √3 = 1.73)

Accepted Answer

Let AB be the vertical pole and P be the point of observation on the ground, where AB = 15 m and ∠APB = 30°.

**Diagram:** [Right triangle with vertical pole AB, horizontal base PB, angle of elevation 30° at P]

In right △ABP,

tan 30° = AB/PB

⟹ 1/√3 = 15/PB

⟹ PB = 15√3 m

Using √3 = 1.73:

⟹ PB = 15 × 1.73 = 25.95 m

∴ The distance of the person from the base of the pole is 25.95 m.

Question 7

A vertical lamp post of height 7 m stands on a horizontal ground. A boy of height 1.4 m walks away from the base of the lamp post at a speed of 1.5 m/s. Find the length of the shadow of the boy after 6 seconds.

Accepted Answer

**Diagram:**

```
L
|\
| \
7m |  \
|   \  1.4m
|    B----T----S
      ←9m→ ←s →
```

*(L = lamp post top, B = base of lamp post, T = top of boy, S = tip of shadow)*

Distance walked by the boy in 6 seconds:

d = 1.5 × 6 = 9 m

Let the length of the shadow of the boy be s m.

The lamp post, the boy, and the ground form two similar triangles.

By AA Similarity Criterion, △LBS ∼ △TPS, where L is the top of the lamp post and T is the top of the boy.

⟹ 7/1.4 = (9 + s)/s

⟹ 5 = (9 + s)/s

⟹ 5s = 9 + s

⟹ 4s = 9

⟹ s = 9/4

∴ The length of the shadow of the boy after 6 seconds = **2.25 m**.

Question 8

A vertical flagpole stands on a horizontal ground. At a point 24 m away from the base of the flagpole, the angle of elevation of the top of the flagpole is 60°. At the same time, a nearby vertical lamp post casts a shadow of 8 m on the ground, and the angle of elevation of the top of the lamp post from the tip of its shadow is 45°. Find the heights of the flagpole and the lamp post. (Use √3 = 1.73)

Accepted Answer

Let the height of the flagpole be h₁ m and the height of the lamp post be h₂ m.

**Height of the flagpole:**

In the right triangle formed by the flagpole, the ground, and the line of sight:

tan 60° = h₁ / 24

⟹ √3 = h₁ / 24

⟹ h₁ = 24√3 = 24 × 1.73

∴ h₁ = 41.52 m

**Height of the lamp post:**

In the right triangle formed by the lamp post, its shadow, and the line of sight from the tip of the shadow:

tan 45° = h₂ / 8

⟹ 1 = h₂ / 8

⟹ h₂ = 8 × 1

∴ h₂ = 8 m

∴ The height of the flagpole is **41.52 m** and the height of the lamp post is **8 m**.

Question 9

A vertical flagpole stands on the ground. From a point on the ground, which is 24 m away from the base of the flagpole, the angle of elevation of the top of the flagpole is 60°. Find the height of the flagpole and the length of its shadow on the ground when the Sun's altitude is 45°.

Accepted Answer

Let the height of the flagpole be *h* m and the length of its shadow be *s* m.

**[Diagram: vertical flagpole AB, point C on ground 24 m from base B, angle of elevation ∠ACB = 60°; shadow BD of length s with Sun's altitude ∠ADB = 45°]**

**Finding the height of the flagpole:**

In right △ABC, using the trigonometric ratio:

tan 60° = AB/BC

⟹ √3 = h/24

∴ h = 24√3 m

**Finding the length of the shadow:**

When the Sun's altitude is 45°, in right △ABD:

tan 45° = AB/BD

⟹ 1 = 24√3/s

∴ s = 24√3 m

∴ The height of the flagpole is **24√3 m** and the length of its shadow on the ground is **24√3 m**.

Question 10

A vertical pole of height 8 m casts a shadow 8√3 m long on the ground. At the same time, a nearby vertical tower casts a shadow 30√3 m long. Find the height of the tower.