From the top of a hill $100\, m$ high,the angles of depression of the top and the bottom of a tower are observed to be $30^{\circ}$ and $45^{\circ}$ respectively. Find the height of the tower in $m$.

As observed from a hill $340 \ m$ high,the angles of depression of two houses situated to the east and west of the hill are found to be $30^{\circ}$ and $60^{\circ}$ respectively. Find the distance between the two houses. (in $m$)

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 $m$)

The lower window of a house is at a height of $2\, m$ above the ground and its upper window is $4\, m$ vertically above the lower window. At a certain instant,the angles of elevation of a balloon from these windows are observed to be $60^{\circ}$ and $30^{\circ}$,respectively. Find the height of the balloon above the ground (in $m$).

From the top of a $7 \,m$ high building,the angle of elevation of the top of a cable tower is $60^{\circ}$ and the angle of depression of its foot is $45^{\circ}$. Determine the height of the tower.

$A$ 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 $a \sec \theta \text{ m}$.