A moving boat is observed from the top of a cliff of 150 m height. The angle of depression of the boat changes from 60$$^\circ$$ to 45$$^\circ$$ in 2 minutes. What is the speed to the boat in metres per hour?

The angle of elevation of a stationary cloud from a point 25m above a lake is 15$$^\circ$$ and the angle of depression of its image in the lake is 45$$^\circ$$. The height of the cloud above the lake level is

The angles of elevation of the top of a tower from the top and foot of a pole are respectively 30$$^\circ$$ and 45$$^\circ$$. If h_T is the height of the tower and h_P is the height of the pole, then which of the following are correct?

1. $${{2{h_P}{h_T}} \over {3 + \sqrt 3 }} = h_P^2$$

2. $${{{h_T} - {h_P}} \over {\sqrt 3 + 1}} = {{{h_P}} \over 2}$$

3. $${{2({h_P} + {h_T})} \over {{h_P}}} = 4 + \sqrt 3 $$

Select the correct answer using the code given below.

From the top of a lighthouse, 100 m high, the angle of depression of a boat is $${\tan ^{ - 1}}\left( {{5 \over {12}}} \right)$$. What is the distance between the boat and the lighthouse?