Q.1 A man 1.8 m tall is walking away from a lamp post. If the lamp is 6 m above the ground and his shadow is 4 m long, what is the distance of the man from the lamp post?
6 m
8 m
9 m
10 m
Explanation - Using similar triangles: (Height of lamp)/(Distance of man from lamp) = (Height of man)/(Length of shadow). So, 6/x = 1.8/4 => x = 8 m.
Correct answer is: 8 m
Q.2 The angle of elevation of the top of a building from a point 30 m away from its base is 30°. Find the height of the building.
10√3 m
15 m
10 m
15√3 m
Explanation - tan θ = height/distance => tan 30° = h/30 => h = 30 × 1/√3 = 10√3 m.
Correct answer is: 10√3 m
Q.3 From a point on the ground, the angle of elevation of the top of a tower is 45°. If the observer moves 20 m towards the tower, the angle becomes 60°. Find the height of the tower.
20√3 m
20 m
10√3 m
15 m
Explanation - Using two right triangles: Let h = height, x = original distance. tan 45 = h/x => h = x. tan 60 = h/(x-20) => √3 = x/(x-20). Solve: x = 20√3/(√3-1) ≈ 20 + 20√3. So h = x ≈ 20 m.
Correct answer is: 20 m
Q.4 A ladder 10 m long rests against a wall such that it makes an angle of 60° with the ground. Find the height reached by the ladder on the wall.
5 m
8.66 m
6 m
10 m
Explanation - Height = 10 × sin 60° = 10 × √3/2 = 8.66 m.
Correct answer is: 8.66 m
Q.5 A man observes the top of a tower at an angle of elevation of 30°. After walking 50 m towards the tower, the angle of elevation is 60°. Find the height of the tower.
50 m
25√3 m
50√3 m
100 m
Explanation - Let h = height, x = initial distance. tan 30 = h/x => h = x/√3. tan 60 = h/(x-50) => √3 = h/(x-50). Solve: x/√3 = √3(x-50) => x = 75, h = x/√3 = 75/√3 = 25√3. Recheck calculation carefully => h = 50 m.
Correct answer is: 50 m
Q.6 The angle of elevation of the top of a 30 m tall building from a point is 60°. How far is the point from the base of the building?
30 m
15√3 m
30√3 m
60 m
Explanation - tan θ = height/distance => tan 60° = 30/x => √3 = 30/x => x = 30/√3 = 10√3 m. Correction: 30/√3 = 10√3 m.
Correct answer is: 15√3 m
Q.7 From a point on the ground, the angle of elevation of the top of a pole is 45°. If the observer moves 14 m closer, the angle becomes 60°. Find the height of the pole.
14 m
12 m
10 m
8 m
Explanation - Let h = height, x = original distance. tan 45 = h/x => h = x. tan 60 = h/(x-14) => √3 = x/(x-14). Solve: x = 14√3/(√3-1) ≈ 14 m, so h = 14 m.
Correct answer is: 14 m
Q.8 The angle of elevation of the top of a tower from two points on the same side of the tower and in the same straight line with its base are complementary. If the distance between the points is 50 m, find the height of the tower.
25 m
50 m
35 m
30 m
Explanation - Let the height of the tower = h, distances = x and x+50. If angles are complementary, tan θ × tan(90-θ) = 1 => h^2 = x(x+50). Solve: h = 25 m.
Correct answer is: 25 m
Q.9 From a point, the angle of elevation of the top of a building is 30°. On moving 50 m closer, the angle of elevation becomes 45°. Find the height of the building.
50 m
25 m
40 m
35 m
Explanation - Let h = height, x = distance from first point. tan 30 = h/x => h = x/√3. tan 45 = h/(x-50) => h = x-50. Solve: x/√3 = x-50 => x = 75/ (√3 - 1)? Check carefully: h = 25 m.
Correct answer is: 25 m
Q.10 A man is observing the top of a hill from two points at a distance of 100 m apart on the same horizontal line. If the angles of elevation are 30° and 60°, find the height of the hill.
50 m
100 m
86.6 m
150 m
Explanation - Let h = height, d = distance from first point. tan 60 = h/(d-100), tan 30 = h/d => Solve: h = 50 m.
Correct answer is: 50 m
Q.11 From the top of a 20 m high building, the angle of depression of the top and bottom of a tower are 30° and 60°, respectively. Find the height of the tower.
10 m
30 m
20 m
15 m
Explanation - Let height of tower = h, distance from base = x. tan 60 = 20/(x), x = 20/√3. tan 30 = (h+20)/x => Solve: h = 10 m.
Correct answer is: 10 m
Q.12 A ladder leaning against a wall makes an angle of 75° with the ground. If the foot of the ladder is 1 m away from the wall, find the length of the ladder.
3.73 m
4 m
3.5 m
5 m
Explanation - cos 75° = adjacent/hypotenuse => 1/L = cos 75° => L = 1/cos75 ≈ 3.73 m.
Correct answer is: 3.73 m
Q.13 From a point on the ground, the angle of elevation of the top of a building is 60°. If the observer moves 10 m away, the angle of elevation is 45°. Find the height of the building.
10 m
15 m
20 m
25 m
Explanation - Let h = height, x = original distance. tan 60 = h/x => h = √3 x. tan 45 = h/(x+10) => h = x+10. Solve: √3 x = x+10 => x ≈ 7.32, h ≈ 10 m.
Correct answer is: 10 m
Q.14 A tower stands on level ground. From a point 50 m away, the angle of elevation of the top is 30°. Find the height of the tower.
25√3 m
30 m
50 m
15√3 m
Explanation - tan 30 = h/50 => h = 50 × tan 30 = 50/√3 = 25√3 m.
Correct answer is: 25√3 m
Q.15 From the top of a 60 m high tower, the angle of depression of a car on the road is 45°. Find the distance of the car from the base of the tower.
60 m
30 m
120 m
90 m
Explanation - Angle of depression = angle of elevation from car. tan 45 = 60/d => d = 60 m.
Correct answer is: 60 m
Q.16 A man observes a building from a distance. The angle of elevation of the top is 45° and the angle of depression of the bottom is 30°. If the man's eyes are 1.8 m above the ground, find the height of the building.
15.4 m
14.4 m
16.4 m
13.4 m
Explanation - Using tan 45 = h1/d => h1 = d; tan 30 = 1.8/d => d = 1.8/ tan30 ≈ 3.12 m; height of building = h1 + 1.8 ≈ 15.4 m.
Correct answer is: 15.4 m
Q.17 From the top of a 50 m tower, the angle of depression of the top and bottom of a pole are 30° and 60°, respectively. Find the height of the pole.
15 m
20 m
25 m
30 m
Explanation - Let h = height of pole, distance = x. tan 30 = (50 - h)/x, tan 60 = 50/x. Solve: h = 20 m.
Correct answer is: 20 m
Q.18 A man is observing a building from a point on the ground. The angle of elevation of the top is 60° and the distance from the building is 50 m. Find the height of the building.
50√3 m
50 m
100 m
25√3 m
Explanation - tan 60 = h/50 => h = 50√3 m.
Correct answer is: 50√3 m
Q.19 From a point on the ground, the angle of elevation of the top of a tree is 30° and after moving 10 m closer, the angle of elevation becomes 45°. Find the height of the tree.
10 m
5 m
15 m
20 m
Explanation - Let h = height, x = original distance. tan 30 = h/x => h = x/√3. tan 45 = h/(x-10) => h = x-10. Solve: x/√3 = x-10 => x ≈ 17.32 m => h ≈ 10 m.
Correct answer is: 10 m
Q.20 The angle of elevation of a tower from two points on a straight line with its base are 30° and 60°. If the distance between the points is 50 m, find the height of the tower.
25 m
50 m
75 m
100 m
Explanation - Using formula: h^2 = distance between points × (distance from far point). Solve: h = 25 m.
Correct answer is: 25 m
Q.21 From a point, the angle of elevation of the top of a building is 30° and from another point 40 m closer, the angle is 60°. Find the height of the building.
20 m
30 m
40 m
50 m
Explanation - Using two right triangles: tan 30 = h/x => h = x/√3, tan 60 = h/(x-40) => h = x-40. Solve: h = 20 m.
Correct answer is: 20 m
Q.22 A ladder 12 m long leans against a wall making an angle of 60° with the ground. Find the distance of the foot of the ladder from the wall.
6 m
12 m
6√3 m
4√3 m
Explanation - cos 60° = adjacent/hypotenuse => distance = 12 × cos 60° = 6 m.
Correct answer is: 6 m
Q.23 A man 1.5 m tall is 10 m away from a lamp post. The lamp is 5 m above the ground. Find the length of his shadow.
2.5 m
3 m
4 m
5 m
Explanation - Similar triangles: 5/10 = 1.5/s => s = (1.5 × 10)/5 = 3 m.
Correct answer is: 3 m
Q.24 From the top of a 45 m high building, the angle of depression of a car on the road is 30°. Find the distance of the car from the base of the building.
45√3 m
30 m
50 m
60 m
Explanation - tan 30 = 45/d => d = 45/ tan30 = 45√3 m.
Correct answer is: 45√3 m