If the angles of elevation of two towers from the middle point of the line joining their feet are $60^\circ$ and $30^\circ$ respectively,then the ratio of their heights is

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.

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

$A$ man is walking towards a vertical pillar in a straight path,at a uniform speed. At a certain point $A$ on the path,he observes that the angle of elevation of the top of the pillar is $30^o$. After walking for $10 \text{ minutes}$ from $A$ in the same direction,at a point $B$,he observes that the angle of elevation of the top of the pillar is $60^o$. Then the time taken (in minutes) by him,from $B$ to reach the pillar,is:

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 ....... $m$.

$A$ $20 \ m$ long tree is broken by the wind such that its top touches the ground at an angle of $30^\circ$. The height from the ground at which the tree is broken is: