## 1. From the top of a tower, the angle of depression of a point on the ground is 30°. If the tower's height is h, the distance of the point from the base of the tower is:

### Answer:

### Explanation:

Let's use tan θ = opposite/adjacent. For angle of depression, tan 30° = h/distance. This gives distance = h√3.

## 2. The angle of elevation of a ladder leaning against a wall is 60° and the foot of the ladder is 2.5 m away from the wall. The length of the ladder is:

### Answer:

### Explanation:

Using tan 60° = height/2.5, height = 2.5√3. Using Pythagoras theorem, ladder length = 5 m.

## 3. From a point on the ground, the angle of elevation to the top of a building is 45°. If the building is 20m tall, the distance of the point from the base of the building is:

### Answer:

### Explanation:

For 45° angle of elevation, height = distance. So, distance = 20 m.

## 4. The shadow of a flagpole is √3 times its height. The angle of elevation of the sun is:

### Answer:

### Explanation:

tan θ = height/shadow = 1/√3, so θ = 30°.

## 5. The angle of depression from the top of a cliff to a boat in the water is 30°. If the cliff is 100m high, how far is the boat from the base of the cliff?

### Answer:

### Explanation:

Using tan 30° = 100/distance, distance = 200 m.

## 6. The angle of elevation of the top of a tower from a point A is 30°. On moving 40m towards the tower, the angle of elevation is 60°. The height of the tower is:

### Answer:

### Explanation:

By applying the tangent formula twice and forming two equations, we can derive the tower's height.

## 7. A man observes the angle of elevation of a bird as 45°. If he walks 10m towards the bird, the angle becomes 60°. The initial distance between the man and the bird is:

### Answer:

### Explanation:

By setting up two tangent relations and solving, we get the initial distance.

## 8. The angle of depression of two ships from the top of a lighthouse are 30° and 45° respectively. If the two ships are in line with the base of the lighthouse, find the ratio of their distances from the lighthouse.

### Answer:

### Explanation:

Using the tangent of angles, we can derive the ratio of distances.

## 9. A tree breaks due to storm and the broken part bends so that the top of the tree touches the ground making an angle of 30°. If the distance between the foot of the tree to the point where the top touches the ground is 10m, the height of the tree is:

### Answer:

### Explanation:

By setting up a relation using tan 30° and the given distance, we can find the height of the tree.

## 10. From the top of a building, the angles of depression of the top and bottom of a tower are 30° and 45° respectively. The height of the building is 30m. The height of the tower is:

### Answer:

### Explanation:

By setting up two tangent relations for top and bottom, we can find the height of the tower.

## 11. A kite is flying at a height of 60m above the ground. The string attached to the kite is temporarily tied to a point on the ground. If the string makes an angle of 45° with the ground, the length of the string is:

### Answer:

### Explanation:

Since tan 45° = 1, height = base. Using Pythagoras theorem, the hypotenuse or the string length = 60√2 m.

## 12. From the top of a tower, the angle of depression of two cars on the opposite sides of the tower are 30° and 45°. If the height of the tower is 60m, the distance between the two cars is:

### Answer:

### Explanation:

Using the tangent of both angles, we can find the distance from the tower to both cars and then find the distance between them.

## 13. The shadow of a tower standing on a level ground is found to be 40m longer when the sun's altitude is 45° than when it is 60°. The height of the tower is:

### Answer:

### Explanation:

By setting up equations using the given conditions and tangent of angles, we can solve for the height of the tower.

## 14. The angle of elevation of a jet plane from a point A on the ground is 60°. After a flight of 15 seconds, the angle of elevation changes to 30°. If the jet plane is flying at a constant height, the speed of the jet plane is:

### Answer:

### Explanation:

We can find the distance traveled using the tangent of both angles, and then divide by the time to get the speed.

## 15. A man on the top of a tower observes a truck at an angle of depression of 30°, which is approaching the tower with a uniform speed. Six seconds later, the angle of depression of the truck is observed to be 45°. The speed of the truck is:

### Answer:

### Explanation:

By determining the change in distance using the tangent of both angles and dividing by the time, we can find the speed.

## 16. An observer 1.5m tall is 28.5m away from a chimney. The angle of elevation from his eye to the top of the chimney is 45°. If the observer’s eye level is 1.2m above the ground, the height of the chimney is:

### Answer:

### Explanation:

We can determine the height difference using tan 45° and add the height of the observer's eye level from the ground.

## 17. A boy observes the angle of elevation of the top of a tower to be 30°. He walks towards it in a straight line and after walking 20m, the angle of elevation becomes 60°. The height of the tower is:

### Answer:

### Explanation:

By setting up and solving two equations using the tangent of the angles, we can determine the height of the tower.

## 18. The angle of elevation of the top of a cliff from a fixed point is 30°. After moving 100m towards the cliff in a straight line, the angle of elevation becomes 45°. The height of the cliff is:

### Answer:

### Explanation:

By using trigonometric ratios in both scenarios, we can set up equations to solve for the height of the cliff.

## 19. A man is watching from the window of a building, the top of a tree which is at some distance from the building. The angle of elevation is found to be 60°. From the bottom of the building, the angle of elevation is found to be 45°. If the building is 30m high, the height of the tree is:

### Answer:

### Explanation:

We can determine the difference in heights using the tangent of both angles and then find the actual height of the tree.

## 20. From the top of a building, the angle of depression of a car on the ground is 45°. If the building is 50m high, the distance of the car from the building is:

### Answer:

### Explanation:

Given the angle and the height, using the tangent of the angle of depression, we can determine the distance of the car from the building.