Question 1

A ladder 20 m long leans against a vertical wall. If the foot of the ladder is 10 m away from the wall, what is the angle the ladder makes with the ground? (Use cos 60° = 0.5)

Accepted Answer

Step 1: Let the angle with the ground be θ. cos θ = adjacent/hypotenuse = 10/20 = 0.5. Step 2: cos θ = 0.5 → θ = 60°. Therefore, the ladder makes an angle of 60° with the ground.

Question 2

A tree casts a shadow of 20 metres on the ground when the angle of elevation of the sun is 60°. What is the height of the tree? (Use √3 = 1.732)

Accepted Answer

Step 1: Use tan(60°) = height/shadow length. Step 2: tan(60°) = √3. Step 3: √3 = height/20. Step 4: height = 20√3 = 20 × 1.732 = 34.64 metres.

Question 3

A man standing 50 metres away from the base of a tower observes the angle of elevation to the top of the tower to be 60°. Find the height of the tower. (Use √3 = 1.732)

Accepted Answer

Step 1: Use the trigonometric ratio tan(θ) = opposite/adjacent. Step 2: tan(60°) = height/50. Step 3: √3 = height/50. Step 4: height = 50√3 = 50 × 1.732 = 86.6 metres.

Question 4

An observer standing 50 metres away from a tree observes the angle of elevation to the top of the tree to be 30°. What is the height of the tree?

Accepted Answer

Step 1: Use tan(angle) = opposite/adjacent. Step 2: tan(30°) = height/50. Step 3: 1/√3 = height/50. Step 4: height = 50/√3 = (50√3)/3 metres. Rationalizing: height ≈ 28.87 metres or (50√3)/3 metres. Therefore, the height is (50√3)/3 metres.

Question 5

A man standing 30 metres away from the base of a tower observes the angle of elevation to the top of the tower to be 60°. Find the height of the tower.

Accepted Answer

Step 1: Use the tangent ratio: tan(angle) = opposite/adjacent. Step 2: tan(60°) = height/30. Step 3: √3 = height/30. Step 4: height = 30√3 metres ≈ 51.96 metres. Therefore, the height of the tower is 30√3 metres.

Question 6

From a point on the ground 30 metres away from a building, the angle of elevation to the top is 45°. What is the height of the building?

Accepted Answer

Step 1: Use tan(45°) = height/distance. Step 2: tan(45°) = 1. Step 3: 1 = height/30. Step 4: height = 30 metres.

Question 7

A ladder leans against a wall. The foot of the ladder is 8 metres from the wall, and the angle between the ladder and the ground is 60°. What is the length of the ladder?

Accepted Answer

Step 1: Use cos(60°) = adjacent/hypotenuse. Step 2: cos(60°) = 8/ladder length. Step 3: 0.5 = 8/ladder length. Step 4: ladder length = 8/0.5 = 16 metres.

Question 8

An observer on the ground sees a bird at an angle of elevation of 30°. The bird is directly above a point 40√3 metres away. What is the height of the bird above the ground? (Use √3 = 1.732)

Accepted Answer

Step 1: Use tan(30°) = height/horizontal distance. Step 2: tan(30°) = 1/√3. Step 3: 1/√3 = height/(40√3). Step 4: height = (40√3)/√3 = 40 metres.

Question 9

From the top of a 15-metre tall building, the angle of depression to a point on the ground is 45°. What is the horizontal distance from the building to that point?

Accepted Answer

Step 1: Angle of depression equals angle of elevation in the right triangle formed. Step 2: Use tan(angle) = opposite/adjacent. Step 3: tan(45°) = height/distance. Step 4: 1 = 15/distance. Step 5: distance = 15 metres. Therefore, the horizontal distance is 15 metres.

Question 10

A man standing 30 metres away from the base of a tower observes the angle of elevation to the top of the tower as 60°. Find the height of the tower. (Use √3 = 1.732)

Accepted Answer

Step 1: Use the tangent ratio in the right triangle formed. tan(60°) = height / distance. Step 2: tan(60°) = √3, so √3 = h / 30. Step 3: h = 30√3 = 30 × 1.732 = 51.96 ≈ 52 metres.

Question 11

A man observes the angle of elevation to the top of a tree as 30° from a point 50 metres away from its base. What is the height of the tree? (Use √3 = 1.732)

Accepted Answer

Step 1: Use tan(30°) = height / distance. Step 2: tan(30°) = 1/√3, so 1/√3 = h / 50. Step 3: h = 50 / √3 = 50 / 1.732 ≈ 28.87 metres, which rounds to approximately 29 metres or 50/√3 metres.

Question 12

From a point on the ground, the angle of elevation to the top of a 20-metre tall building is 45°. How far is the point from the base of the building?

Accepted Answer

Step 1: Use tan(angle) = opposite/adjacent. Step 2: tan(45°) = height/distance. Step 3: 1 = 20/distance. Step 4: distance = 20 metres. Therefore, the point is 20 metres away from the base of the building.

Question 13

A ladder leans against a wall such that it makes an angle of 30° with the ground. If the ladder is 10 metres long, what is the height at which the ladder touches the wall?

Accepted Answer

Step 1: Use sin(angle) = opposite/hypotenuse. Step 2: sin(30°) = height/ladder length. Step 3: 1/2 = height/10. Step 4: height = 5 metres. Therefore, the ladder touches the wall at a height of 5 metres.

Question 14

From the top of a cliff 80 metres high, the angle of depression to a boat on the water is 45°. How far is the boat from the base of the cliff?

Accepted Answer

Step 1: Use tan(45°) = height/horizontal distance. Step 2: tan(45°) = 1. Step 3: 1 = 80/distance. Step 4: distance = 80 metres.

Question 15

From the top of a cliff 80 metres high, the angle of depression to a boat on the water is 45°. How far is the boat from the base of the cliff?

Accepted Answer

Step 1: In the angle of depression problem, the angle of depression equals the angle of elevation from the boat's perspective. Step 2: We have a right triangle where height (opposite) = 80 m and angle = 45°. Step 3: Using tan(45°) = height/distance, we get 1 = 80/distance, so distance = 80 metres.

Question 16

Two buildings stand on level ground. From the top of the first building (height 20 m), the angle of depression to the base of the second building is 30°. If the buildings are on the same horizontal line, find the horizontal distance between them. [Use √3 ≈ 1.732]

Accepted Answer

Step 1: Height of first building = 20 metres. Angle of depression = 30°. Step 2: Angle of depression equals angle of elevation from the opposite point. So tan(30°) = height/distance. Step 3: 1/√3 = 20/distance. Step 4: distance = 20√3 = 20 × 1.732 = 34.64 metres.

Question 17

A man standing 100 m away from the base of a tower observes the angle of elevation to the top of the tower as 30°. Find the height of the tower (in metres).

Accepted Answer

Step 1: We have a right triangle where the horizontal distance (adjacent) = 100 m and angle of elevation = 30°. Step 2: Using tan(30°) = height/distance, we get height = 100 × tan(30°) = 100 × (1/√3) = 100/√3 = 100√3/3 ≈ 57.74 m. Step 3: Rationalizing: 100/√3 × √3/√3 = 100√3/3 m. For practical purposes, this equals approximately 57.74 m, but the exact answer is 100/√3 m or 100√3/3 m.

Question 18

From a point on the ground 80 m away from a building, the angle of elevation to the top is 45°. What is the height of the building?

Accepted Answer

Step 1: In this right triangle, horizontal distance = 80 m and angle of elevation = 45°. Step 2: Using tan(45°) = height/distance, and since tan(45°) = 1, we get height = 80 × 1 = 80 m. Step 3: Therefore, the height of the building is 80 m.

Question 19

An observer at height 1.5 m above ground level observes the angle of depression to a point on the ground as 30°. How far is the point from the base of the observer's position (horizontal distance)?

Accepted Answer

Step 1: The observer is at height 1.5 m, and the angle of depression to a point on ground is 30°. Step 2: In the right triangle formed, the vertical height (opposite) = 1.5 m and angle of depression = 30°. Step 3: Using tan(30°) = opposite/adjacent, we get tan(30°) = 1.5/distance. Step 4: 1/√3 = 1.5/distance, so distance = 1.5√3 m ≈ 2.598 m.

Question 20

From the top of a cliff 120 m high, the angle of depression to a boat on the water is 45°. How far is the boat from the base of the cliff (horizontal distance)?

Accepted Answer

Step 1: The cliff height (vertical distance) = 120 m. The angle of depression = 45°. Step 2: In the right triangle, tan(angle of depression) = height/horizontal distance. Step 3: tan(45°) = 120/distance. Step 4: Since tan(45°) = 1, we get 1 = 120/distance, so distance = 120 m. Step 5: The boat is 120 m away from the base of the cliff.

Question 21

A person standing on the ground observes the angle of elevation to the top of a building as 45°. After walking 30 metres closer to the building, the angle of elevation becomes 60°. Find the height of the building.

Accepted Answer

Step 1: Let the initial distance from the building = x metres, and height = h metres. Step 2: From the first position: tan(45°) = h/x, so h = x (since tan(45°) = 1). Step 3: From the second position (30 m closer): tan(60°) = h/(x-30), so h = (x-30)√3. Step 4: Equating: x = (x-30)√3. Step 5: x = x√3 - 30√3, so x(√3 - 1) = 30√3. Step 6: x = 30√3/(√3 - 1). Step 7: Rationalizing: x = 30√3(√3 + 1)/[(√3 - 1)(√3 + 1)] = 30√3(√3 + 1)/(3 - 1) = 30√3(√3 + 1)/2 = 15√3(√3 + 1) = 15(3 + √3) = 45 + 15√3. Step 8: h = x = 45 + 15√3 metres.

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