$A$ $20 \ m$ 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:

$A$ tower $T_1$ of height $60 \, m$ is located exactly opposite to a tower $T_2$ of height $80 \, m$ on a straight road. From the top of $T_1$,if the angle of depression of the foot of $T_2$ is twice the angle of elevation of the top of $T_2$,then the width (in $m$) of the road between the feet of the towers $T_1$ and $T_2$ is

From a point on the level ground, the angle of elevation of the top of a pole is $30^{\circ}$. On moving $20 \ m$ nearer to the pole, the angle of elevation becomes $45^{\circ}$. The height of the pole (in metres) is:

The angle of elevation of the top of a tower from a point $A$ due south of the tower is $\alpha$ and from a point $B$ due east of the tower is $\beta$. If $AB = d$,then the height of the tower is

For a man,the angle of elevation of the highest point of the temple situated east of him is $60^\circ$. On walking $240 \ m$ to the north,the angle of elevation is reduced to $30^\circ$. The height of the temple 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