1. Durga was standing on a railway overbridge that is 5√3 meters high. She observed the engine of a moving passenger train at a depression angle of 30° on one side of the bridge. Two seconds later, she saw the same engine at a depression angle of 60° on the other side of the bridge. Durga's position was vertically above the railway track, which is assumed to be a straight line. Find the speed of the train.

2. Standing on a railway overbridge 5√3 meters high, Amitadi initially saw the engine of a train on this side of the bridge at a depression angle of 30°. After 2 seconds, she saw the engine on the other side of the bridge at a depression angle of 45°. Calculate the speed of the train in meters per second.

3. Suppose a passenger on an airplane, at a certain moment, sees Howrah Station on one side and Shaheed Minar on the exact opposite side, with angles of depression of 60° and 30° respectively. At that time, if the airplane is flying at a height of \(545\sqrt{3}\) meters, then determine the distance between Howrah Station and Shaheed Minar.

4. There is a bridge perpendicular to the bank of a river. From one end of the bridge on the bank, if a person walks a certain distance along the riverbank, the other end of the bridge is seen at an angle of 45°. Walking 400 meters farther along the bank, the same end of the bridge is then seen at an angle of 30°. Find the length of the bridge.

5. A person first observes a bird flying at an angle of elevation of 60° towards the north, and after 5 minutes, sees it at an angle of elevation of 30° towards the south. If the bird is flying along the same straight line at a height of \(120\sqrt{3}\) meters, determine its speed in kilometers per hour.

6. A passenger on an airplane once saw Howrah Station on one side and Shaheed Minar on the opposite side at angles of depression of 60° and 30°, respectively. At that moment, if the airplane was flying at a height of \(545\sqrt{3}\) meters, what is the distance between Howrah Station and Shaheed Minar?

7. A 5-meter-high house is located on one side of a park. From both the roof and the base of the house, the base and the top of a palm tree on the opposite side of the park are seen at angles of depression and elevation of 30° and 60°, respectively. What is the height of the palm tree? What is the distance between the house and the palm tree?

8. A person observes an aircraft from a certain location at an angle of elevation of 60°. After 15 seconds, from the same location, he sees the aircraft again at an angle of elevation of 30°. If the aircraft is flying along the same straight line at a height of \(1500\sqrt{3}\) meters, then what is its speed?

9. Here's the full translation of your math problem into English: > The height of one building forms a right angle with the window of another building situated directly across the street. From that window, the angle of elevation to the top of the first building is 30°. > If the width of the street is \(4\sqrt{3}\) meters, find: > (1) the height of the first building, and > (2) how high the window of the second building is above the street level.

10. Standing in the middle of the field, Habu sees a flying bird first at an elevation angle of 30° to the north, and after 2 minutes, at an elevation angle of 60° to the south. If the bird is flying along a straight path at a constant height of \(50\sqrt{3}\) meters, then what is its speed?

11. The deck of a ship is 10 meters above sea level. Standing on the deck, Apu observes the top of a lighthouse at an angle of elevation of 60°, and its base at an angle of depression of 30°. Find the distance from the ship to the lighthouse and the height of the lighthouse.

12. Standing in an open field, Habul sees a flying bird at an angle of elevation of 30° towards the north, and after 2 minutes, he sees it at an angle of elevation of 60° towards the south. If the bird is flying along a straight horizontal path at a height of \(50\sqrt{3}\) meters, then calculate the speed of the bird.

13. A bird was flying from north to south along a path parallel to the ground at a height of 200 meters. Standing in the middle of a field, Payel first saw the bird at an angle of 30° to the north. Three minutes later, she saw it again at an angle of 45° to the south. Determine the speed of the bird.

14. From the roof of a 15-meter-high house located at one end of a park, the base and top of a brick kiln chimney at the other end of the park are seen at angles of depression and elevation of 30° and 60°, respectively. Determine the height of the brick kiln chimney and the distance between the kiln and the house.

15. The perimeters of two similar triangles are 20 cm and 16 cm respectively. If one side of the first triangle is 9 cm, what is the length of the corresponding side of the second triangle?

16. If the angle of elevation to the top of a monument from the roof of a five-story building (18 meters high) is 45°, and the angle of depression to the base of the monument is 60°, then determine the height of the monument. (√3 = 1.732)

17. “The perimeters of two similar triangles are 20 cm and 16 cm respectively. If the length of one side of the first triangle is 9 cm, what is the length of the corresponding side of the second triangle.”

18. “The perimeters of two similar triangles are 24 cm and 16 cm respectively. If one side of the second triangle is 6 cm, what will be the length of the corresponding side of the first triangle.”

19. Standing in the middle of a field, Mohit saw a flying bird—first to the north at an elevation angle of 30°, and then 2 minutes later to the south at an elevation angle of 60°. If the bird is flying along the same straight path at a height of 50√3 meters, calculate its speed in kilometers per hour.

20. A person in an airship sees Howrah Station and, directly opposite, the Shaheed Minar at angles of depression of 60° and 30°, respectively. If the airship is at a height of \( 545\sqrt{3} \) meters, find the distance between Howrah Station and Shaheed Minar.

21. If from the roof of a 5-storey building that is 18 meters high, the angle of elevation to the top of a monument is 45°, and the angle of depression to the base of the monument is 60°, then calculate the height of the monument. (Take √3 ≈ 1.732) Let me know if you'd like a step-by-step solution as well.

22. Standing in the middle of a field, Mohit saw a flying bird—first to the north at an angle of elevation of 30°, and 2 minutes later to the south at an angle of elevation of 60°. If the bird is flying along the same straight line at a height of 50√3 meters, calculate its speed.

23. “The perimeters of two similar triangles are 20 cm and 16 cm respectively. If the length of one side of the first triangle is 9 cm, calculate the length of the corresponding side of the second triangle.”

24. If from the roof of a 5-storey building that is 18 meters high, the angle of elevation to the top of a monument is 45° and the angle of depression to the base of the monument is 60°, then calculate and write the height of the monument. [Take \(\sqrt{3} \approx 1.732\)]

25. A bird was flying from north to south along a path parallel to the ground at a height of 200 meters. Standing at the center of a field, Sushobhan first saw the bird at an angle of elevation of 30° to the north. Three minutes later, he again saw it at an angle of elevation of 45° to the south. Calculate and write the bird’s speed in kilometers per hour, rounded to the nearest whole number. [Take \(\sqrt{3} \approx 1.732\)]

26. In our neighborhood, there are two houses on opposite sides of the road facing each other. If the base of a ladder is placed 6 meters away from the foot of the wall of the first house and the ladder is leaned against the wall, it makes an angle of 30° with the horizontal. But if the same ladder is placed in the same position and leaned against the wall of the second house, it makes an angle of 60° with the horizontal. (i) Find the length of the ladder. (ii) Calculate how far the foot of the ladder is from the base of the second house. (iii) Find the width of the road. (iv) Determine the height at which the top of the ladder touches the second house.

27. Standing in the middle of a field, Mohit first sees a flying bird at an angle of elevation of 30° to the north. Two minutes later, he sees the same bird at an angle of elevation of 60° to the south. If the bird is flying along a straight line at a height of \(50\sqrt{3}\) meters, determine its speed in kilometers per hour.

28. From the roof of a 15-meter-high house situated at one end of a park, the foot and the top of a brick kiln chimney located at the other end of the park are seen at angles of depression and elevation of 30° and 60°, respectively. Find the height of the chimney and the distance between the brick kiln and the house.

29. A bird was flying from north to south along a straight path parallel to the ground, at a height of 200 meters. Standing in the middle of a field, Sushovan first saw the bird at an angle of elevation of 30° towards the north. After 3 minutes, he saw it again at an angle of elevation of 45° towards the south. Find the bird’s speed in kilometers per hour, rounded to the nearest whole number. [Given: √3 ≈ 1.732]

30. Standing in the middle of the field, Rohit first saw a flying bird at an elevation angle of 30° towards the north, and then 2 minutes later, saw it at an angle of 60° towards the south. If the bird is flying along a straight path at a constant height of 50√3 meters, what is the bird’s speed in kilometers per hour?