As observed from the top of a $75 \, m$ high lighthouse from the sea-level,the angles of depression of two ships are $30^{\circ}$ and $45^{\circ}$. If one ship is exactly behind the other on the same side of the lighthouse,find the distance between the two ships.

$A$ 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 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 $\sqrt{3} = 1.732$)

$A$ straight highway leads to the foot of a tower. $A$ man standing at the top of the tower observes a car at an angle of depression of $30^{\circ}$,which is approaching the foot of the tower with a uniform speed. Six seconds later,the angle of depression of the car is found to be $60^{\circ}$. Find the time taken by the car to reach the foot of the tower from this point (in $seconds$).

$A$ $1.5 \, m$ tall boy is standing at some distance from a $30 \, m$ tall building. The angle of elevation from his eyes to the top of the building increases from $30^{\circ}$ to $60^{\circ}$ as he walks towards the building. Find the distance he walked towards the building.

From a point on a bridge across a river,the angles of depression of the banks on opposite sides of the river are $30^{\circ}$ and $45^{\circ}$,respectively. If the bridge is at a height of $3 \, m$ from the banks,find the width of the river.