When the angle of elevation of the sun is $30^{\circ}$,the length of the shadow of a tower is $90\, m$. Find the height of the tower. (in $m$)

Write 'True' or 'False' and justify your answer.
The angle of elevation of the top of a tower is $30^{\circ}$. If the height of the tower is doubled,then the angle of elevation of its top will also be doubled.

$A$ 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 $min$)

Write 'True' or 'False' and justify your answer.
The value of $\tan \theta$ (where $\theta < 90^{\circ}$) increases as $\theta$ increases.

The angle of depression of the base of a tower from the top of a cliff is $60^{\circ}$ and the angle of elevation of the top of the tower from the base of the cliff is $30^{\circ}$. If the height of the tower is $50 \, m$,find the height of the cliff in meters.

$A$ vertical tower stands on a horizontal plane and is surmounted by a vertical flag staff of height $h$. At a point on the plane,the angles of elevation of the bottom and the top of the flag staff are $\alpha$ and $\beta$,respectively. Prove that the height of the tower is $\left(\frac{h \tan \alpha}{\tan \beta-\tan \alpha}\right)$.