$A$ boat is sailing with constant speed towards a man sitting on a tree on the river bank. At some moment,the man measures the angle of depression of the boat as $30^{\circ}$. $10$ minutes later,this angle is measured as $60^{\circ}$. How much longer will the boat now take to reach the bank? (in $min$)

The distance between two poles of equal height is $200 \, m$. From a point situated on the line segment joining their bases,the angles of elevation of their tops are found to be $60^{\circ}$ and $30^{\circ}$. Find the height of the pole in $m$.

In right-angled $\Delta ABC$,$m\angle B = 90^\circ$. If $AC = 20$ and $m\angle C = 30^\circ$,then $BC = \ldots$

$A$ ladder rests against a vertical wall at an inclination $\alpha$ to the horizontal. Its foot is pulled away from the wall through a distance $p$ so that its upper end slides a distance $q$ down the wall and then the ladder makes an angle $\beta$ to the horizontal.
Show that $\frac{p}{q}=\frac{\cos \beta-\cos \alpha}{\sin \alpha-\sin \beta}$

When the angle of elevation of the sun increases from $30^{\circ}$ to $60^{\circ},$ the shadow of a tower decreases by $50\, m .$ Find the height of the tower in $m$.

$A$ pole $6\, m$ high casts a shadow $2 \sqrt{3}\, m$ long on the ground. Then,the Sun's elevation is: (in $^{\circ}$)