If the angles of depression of the top and bottom of a short building from the top of a tall building are $30^{\circ}$ and $60^{\circ}$ respectively, then the ratio of the heights of the short and tall buildings 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 the top of a vertical tower from a point $A$,due east of it,is $45^\circ$. The angle of elevation of the top of the same tower from a point $B$,due south of $A$,is $30^\circ$. If the distance between $A$ and $B$ is $54\sqrt{2} \, \text{m}$,then the height of the tower (in metres) is:

The shadow of a tower standing on a level plane is found to be $100 \ m$ longer when the sun's angle of elevation is $30^{\circ}$ than when it is $45^{\circ}$. The height of the tower is:

If a flagstaff of $6 \ m$ high placed on the top of a tower throws a shadow of $2\sqrt{3} \ m$ along the ground,then the angle (in degrees) that the sun makes with the ground is.......$^o$

$ABCD$ is a rectangular field. $A$ vertical lamp post of height $12 \ m$ stands at the corner $A$. If the angle of elevation of its top from $B$ is $60^{\circ}$ and from $C$ is $45^{\circ}$,then the area of the field is