On walking for search of a ball $x$ metres on a hill making an angle of $30^{\circ}$ with the ground,one can reach a height of $y$ metres from the ground. Then $\ldots \ldots \ldots . . .$

The angle of elevation of the top of the temple on the bank of the river from a point exactly opposite to the temple on the other bank of the river is $30^{\circ}$. If the height of the temple is $20 \ m$,then the breadth of the river is $\ldots \ldots \ldots \ldots \ m$.

$A$ $15\ m$ high eucalyptus tree standing erect on the ground breaks at the height of $5\ m$ from the ground. The broken part bends such that the top of the tree touches the ground. Find the angle made by the broken part of the tree with the ground. (in $^{\circ}$)

From a point $A$,$h \text{ m}$ above the ground level,the angle of elevation of the top of a tower is $\alpha$ and the angle of depression of the base of the tower is $\beta$. Prove that the height of the tower is $\frac{h(\tan \alpha + \tan \beta)}{\tan \beta} \text{ m}$.

Write 'True' or 'False' and justify your answer.
If the height of a tower and the distance of the point of observation from its foot,both,are increased by $10 \%$,then the angle of elevation of its top remains unchanged.

The height of a tower is $30 \ m$. Find the length of the shadow of the tower,when the angle of elevation of the sun is $45^\circ$. (in $m$)