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

Let $AB$ and $PQ$ be two vertical poles,$160 \ m$ apart from each other. Let $C$ be the middle point of $B$ and $Q$,which are the feet of these two poles. Let $\frac{\pi}{8}$ and $\theta$ be the angles of elevation from $C$ to $P$ and $A$,respectively. If the height of pole $PQ$ is twice the height of pole $AB$,then $\tan^{2} \theta$ is equal to

Two vertical poles of heights $20 \ m$ and $80 \ m$ stand apart on a horizontal plane. The height (in meters) of the point of intersection of the lines joining the top of each pole to the foot of the other,from this horizontal plane is:

The angles of elevation of the top of a tower from three collinear points $A, B$,and $C$ on a road leading to the foot of the tower are $30^{\circ}, 45^{\circ}$,and $60^{\circ}$ respectively. The ratio of $AB$ to $BC$ is

$A$ flag-staff $5 \ m$ high stands on a building of height $25 \ m$. An observer at a height of $30 \ m$ observes that the flag-staff and the building subtend equal angles. The distance of the observer from the top of the flag-staff is:

An observer on the top of a tree finds the angle of depression of a car moving towards the tree to be $30^\circ$. After $3 \text{ minutes}$,this angle becomes $60^\circ$. After how much more time will the car reach the tree? (in minutes)