As observed from the top of a tower $30 \, m$ high,the angle of depression of a stone on the ground is found to be $45^{\circ}$. Find the distance between the stone and the tower (in $m$).

The angles of depression of two cars parked in the same direction of a $50\, m$ high tower are found to be $30^{\circ}$ and $60^{\circ}$. Find the distance between these two cars. (in $, m$)

From a point $P$ on the ground,the angle of elevation of the top of a $10 \, m$ tall building is $30^{\circ}$. $A$ flag is hoisted at the top of the building and the angle of elevation of the top of the flagstaff from $P$ is $45^{\circ}$. Find the length of the flagstaff and the distance of the building from the point $P$. (You may take $\sqrt{3} = 1.732$)

The angle of elevation of the top of a vertical pole as observed from a point $20\, m$ away from the base of the pole is $60^\circ$. Find the height of the pole. (in $m$)

$A$ contractor plans to install two slides for the children to play in a park. For the children below the age of $5$ years,she prefers to have a slide whose top is at a height of $1.5 \, m$,and is inclined at an angle of $30^{\circ}$ to the ground,whereas for elder children,she wants to have a steep slide at a height of $3 \, m$,and inclined at an angle of $60^{\circ}$ to the ground. What should be the length of the slide in each case?

From the top of a tower,the angles of depression of points $A$ and $B$ in the same direction of the tower are found to be $\theta$ and $(90^\circ - \theta)$ respectively. If $B$ is nearer the tower than $A$ and $AB = a$,then prove that the height of the tower is $\frac{a \tan \theta}{1 - \tan^2 \theta}$.