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.

$A$ 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 $seconds$).

The angle of elevation of the top of a building from the foot of the tower is $30^{\circ}$ and the angle of elevation of the top of the tower from the foot of the building is $60^{\circ}$. If the tower is $50\, m$ high,find the height of the building.

Two poles of equal heights are standing opposite each other on either side of the road,which is $80\, m$ wide. From a point between them on the road,the angles of elevation of the top of the poles are $60^{\circ}$ and $30^{\circ},$ respectively. Find the height of the poles and the distances of the point from the poles.

$A$ tree breaks due to a storm and the broken part bends so that the top of the tree touches the ground,making an angle of $30^{\circ}$ with it. The distance between the foot of the tree and the point where the top touches the ground is $8\, m$. Find the height of the tree.

$A$ 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?