Consider a triangular plot $ABC$ with sides $AB = 7 \ m$,$BC = 5 \ m$,and $CA = 6 \ m$. $A$ vertical lamp-post at the midpoint $D$ of $AC$ subtends an angle $30^o$ at $B$. The height (in $m$) of the lamp-post is

If the angles of depression of the top and bottom of a short building from the top of a tall building are $30^{\circ}$ and $60^{\circ}$ respectively, then the ratio of the heights of the short and tall buildings is

The angle of elevation of the top of a vertical tower from a point $A$,due east of it,is $45^\circ$. The angle of elevation of the top of the same tower from a point $B$,due south of $A$,is $30^\circ$. If the distance between $A$ and $B$ is $54\sqrt{2} \, \text{m}$,then the height of the tower (in metres) 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)

Let $10$ vertical poles standing at equal distances on a straight line,subtend the same angle of elevation $\alpha$ at a point $O$ on this line and all the poles are on the same side of $O$. If the height of the longest pole is $h$ and the distance of the foot of the smallest pole from $O$ is $a$,then the distance between two consecutive poles is:

$ABC$ is a triangular park with $AB = AC = 100 \text{ metres}$. $A$ vertical tower of height $h$ is situated at the mid-point $P$ of $BC$. If the angles of elevation of the top of the tower $Q$ at $A$ and $B$ are $\cot^{-1}(3\sqrt{2})$ and $\csc^{-1}(2\sqrt{2})$ respectively,then the height of the tower (in metres) is