The tops of two poles of height $20 \ m$ and $14 \ m$ are connected by a wire. If the wire makes an angle $30^{\circ}$ with the horizontal,then the length of the wire is $... \ m$.

$A$ house of height $100 \ m$ subtends a right angle at the window of an opposite house. If the height of the window is $64 \ m$,then the distance between the two houses is......$m$.

The angle of elevation of the top of a tower from a point $A$ due north of it is $\alpha$ and from a point $B$ at a distance of $9$ units due west of $A$ is $\cos^{-1}\left(\frac{3}{\sqrt{13}}\right)$. If the distance of the point $B$ from the tower is $15$ units,then $\cot \alpha$ is equal to.

The angle of elevation of the top of a tower from a point $20 \ m$ away from its base is $45^\circ$. The height of the tower is ....... $m$.

The angle of elevation of an object on a hill is observed from a certain point in the horizontal plane through its base to be $30^{\circ}$. After walking $120 \ m$ towards it on level ground, the angle of elevation is found to be $60^{\circ}$. Then the height of the object (in metres) is:

The angle of elevation of the top of a vertical tower from a point $P$ on the horizontal ground was observed to be $\alpha$. After moving a distance $2 \ m$ from $P$ towards the foot of the tower,the angle of elevation changes to $\beta$. Then the height (in metres) of the tower is