$A$ $15\ m$ high eucalyptus tree standing erect on the ground breaks at the height of $5\ m$ from the ground. The broken part bends such that the top of the tree touches the ground. Find the angle made by the broken part of the tree with the ground. (in $^{\circ}$)

$A$ cable tied to the top of an electric pole is affixed at a point on the ground $a \text{ m}$ away from the pole. If the cable makes an angle $\theta$ with the ground,then prove that the height of the pole is $a \tan \theta \text{ m}$ and the length of the wire is $a \sec \theta \text{ m}$.

The angle of elevation of the top of a hill from a point on the ground is $30^\circ$. After walking $30 \, m$ towards the hill,the angle of elevation becomes $45^\circ$. What is the height of the hill? (in $m$)

$A$ $1.2 \, m$ tall girl spots a balloon moving with the wind in a horizontal line at a height of $88.2 \, m$ from the ground. The angle of elevation of the balloon from the eyes of the girl at any instant is $60^{\circ}$. After some time,the angle of elevation reduces to $30^{\circ}$. Find the distance travelled by the balloon during this interval.

As observed from the top of a $510 \ m$ high tower,the angles of depression of two houses situated in the east and west of the tower are found to be $30^{\circ}$ and $60^{\circ}$ respectively. Find the distance between the two houses (in $m$).

During a storm,a $21 \,m$ high tree breaks but the broken part remains attached to the trunk and makes an angle of $30^{\circ}$ with the ground. At what height from the ground is the tree broken? (in $m$)