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

The upper part of a tree broken by the wind makes an angle of $30^\circ$ with the ground. The distance from the root to the point where the top of the tree touches the ground is $10 \, m$. What was the original height of the tree in meters?

The angle of elevation of a tower from a point $A$ due south of it is $30^\circ$ and from a point $B$ due west of it is $45^\circ$. If the height of the tower is $100 \ m$,then $AB = \dots \ m$.

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 pillars of equal height stand on either side of a roadway which is $60 \, m$ wide. At a point in the roadway between the pillars,the angles of elevation of the top of the pillars are $60^\circ$ and $30^\circ$. The height of the pillars is

From the top of a lighthouse $30 \ m$ high with its base at sea level,the angle of depression of a boat is $15^\circ$. The distance of the boat from the foot of the lighthouse is: