The upper part of a tree broken by the wind makes an angle of $30^{\circ}$ with the ground and the distance from the root to the point where the top of the tree touches the ground is $10 \ m$. What was the height of the tree?

Two posts are $2 \text{ m}$ apart. Both posts are on the same side of a tree. If the angles of depression of these posts when observed from the top of the tree are $45^{\circ}$ and $60^{\circ}$ respectively,then the height of the tree is:

The length of a shadow of a vertical tower is $\frac{1}{\sqrt{3}}$ times its height. The angle of elevation of the Sun is (in $^{\circ}$)

$ABCD$ is a rectangle where the ratio of the lengths of $AB$ and $BC$ is $3:2$. If $P$ is the midpoint of $AB$,then the value of $\sin(\angle CPB)$ is

$A$ man on the top of a vertical observation tower observes a car moving at a uniform speed coming directly towards it. If it takes $12$ minutes for the angle of depression to change from $30^{\circ}$ to $45^{\circ},$ how soon after this will the car reach the observation tower?

An aeroplane flying at a height of $5000 \; m$ from the ground passes vertically above another aeroplane at an instant. At that instant,the angles of elevation of the two aeroplanes from the same point on the ground are $60^{\circ}$ and $45^{\circ}$ respectively. The vertical distance between the aeroplanes at that instant is: