$A$ 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 angles of elevation of the top of a tower from two points at a distance of $4 \,m$ and $9 \,m$ from the base of the tower and in the same straight line with it are complementary. Prove that the height of the tower is $6 \,m$.

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.

The shadow of a tower standing on a level ground is found to be $40 \, m$ longer when the Sun's altitude is $30^{\circ}$ than when it is $60^{\circ} .$ Find the height of the tower.

$A$ 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?

$A$ $1.2\, m$ 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 $30^{\circ}$ (see $Fig.$). Find the distance travelled by the balloon during the interval.