$A$ tower stands vertically on the ground. The angle of elevation of the top of the tower from a point $100 \, m$ away from the base of the tower is $60^\circ$. The height of the tower is $\ldots \ldots \ldots \ldots \, m$.

The height of a building is $11 \, m$ and the height of a pole is $8 \, m$. Watching from the midpoint of the line segment joining their bases,the angle of elevation of the top of the pole is $\alpha$ and that of the top of the building is $\beta$. Then,$\ldots \ldots \ldots$ holds true.

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.

The angle of elevation of the top of a tower $30 \, m$ high from the foot of another tower in the same plane is $60^{\circ}$ and the angle of elevation of the top of the second tower from the foot of the first tower is $30^{\circ}$. Find the distance between the two towers and also the height of the other tower.

Two poles of equal heights are standing opposite each other on either side of the road,which is $80\, m$ wide. From a point between them on the road,the angles of elevation of the top of the poles are $60^{\circ}$ and $30^{\circ},$ respectively. Find the height of the poles and the distances of the point from the poles.

From the top of a tower,the angles of depression of two vehicles on the same side of the tower are found to be $\alpha$ and $\beta$ $(\alpha > \beta)$. If the distance between the vehicles is $b$,show that the height of the tower is $\frac{b \tan \alpha \tan \beta}{\tan \alpha - \tan \beta}$.