$A$ person observes the angle of elevation of a building as $30^\circ$. The person proceeds towards the building with a speed of $25(\sqrt{3} - 1) \, m/hour$. After $2 \, hours$,he observes the angle of elevation as $45^\circ$. The height of the building (in meters) is:

$A$ man is observing,from the top of a tower,a boat speeding towards the tower from a certain point $A$,with uniform speed. At that point,the angle of depression of the boat from the man's eye is $30^{\circ}$ (ignore the man's height). After sailing for $20 \text{ seconds}$ towards the base of the tower (which is at the level of water),the boat reaches a point $B$,where the angle of depression is $45^{\circ}$. Then,the time taken (in seconds) by the boat from $B$ to reach the base of the tower is:

If the angle of elevation of the top of a tower at a distance of $500 \, m$ from its foot is $30^\circ$,then the height of the tower is:

$A$ tree is broken by wind,its upper part touches the ground at a point $10 \ m$ from the foot of the tree and makes an angle of $45^\circ$ with the ground. The total length of the tree is:

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

Two parallel towers $A$ and $B$ of different heights are at some distance $d$ on the same level ground. If the angle of elevation of a point $P$ at $20\,m$ height on tower $B$ from a point $Q$ at $10\,m$ height on tower $A$ is $\theta$,and this angle is equal to half the angle of elevation of point $R$ at $50\,m$ height on tower $A$ from point $P$ on tower $B$,then $\theta$ is equal to....$^o$