$A$ pole stands vertically inside a triangular park $\Delta ABC$. Let the angle of elevation of the top of the pole from each corner of the park be $\frac{\pi}{3}$. If the radius of the circumcircle of $\Delta ABC$ is $2$,then the height of the pole is equal to:

If the angle of elevation of a cloud from a point $P$ which is $25 \, m$ above a lake is $30^o$ and the angle of depression of the reflection of the cloud in the lake from $P$ is $60^o$,then the height of the cloud (in meters) from the surface of the lake is:

The angle of elevation of a stationary cloud from a point $2500 \, m$ above a lake is $15^\circ$ and the angle of depression of its reflection in the lake is $45^\circ$. The height of the cloud above the lake level is

Two men are on the opposite sides of a tower. They measure the angles of elevation of the top of the tower as $45^{\circ}$ and $30^{\circ}$ respectively. If the height of the tower is $40 \, m,$ find the distance between the men.

If the angle of depression of a point $A$ on the ground from the top of a tower is $30^\circ$,then the angle of elevation of the top of the tower from the point $A$ will be .............. $^\circ $.

$A$ person observes the top of a tower from a point $A$ on the ground. The angle of elevation of the tower from this point is $60^{\circ}$. He moves $60 \ m$ in the direction perpendicular to the line joining $A$ and the base of the tower to reach point $C$. The angle of elevation of the tower from point $C$ is $45^{\circ}$. Then,the height of the tower (in metres) is