$A$ pole stands slanting on the ground making an angle of $60^{\circ}$ with the ground. At high noon,the length of the shadow of the pole is $3 \, m$. Find the length of the pole 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 $m$).

$A$ ladder leans against a wall. It makes an angle of $45^{\circ}$ with the ground and reaches a height of $10 \, m$ on the wall. Find the length of the ladder and the distance between the lower end of the ladder and the base of the wall.

Watching from a point $A$ on the ground $x \ m$ away from the base of the building. The height of the building is $y$ and the angle of elevation of the top of the building is found to be $25^\circ$. Then,$\ldots \ldots \ldots \ldots .$

From a point $h$ metres above the water level of a lake,the angle of elevation of the top of a palace is found to be $\alpha$ and the angle of depression of the image of the top of the palace observed in the lake is found to be $\beta$. Find the height of the palace.

The angle of elevation of the top of a tower from two points at distances $s$ and $t$ from its foot are complementary. Prove that the height of the tower is $\sqrt{s t}$.