The shadow of a tower of height $(1 + \sqrt{3}) \text{ m}$ standing on the ground is found to be $2 \text{ m}$ longer when the sun's elevation is $30^{\circ}$ than when the sun's elevation was $...^{\circ}$.

$P$ is a point on the segment joining the feet of two vertical poles of heights $a$ and $b$. The angles of elevation of the tops of the poles from $P$ are $45^{\circ}$ each. Then, the square of the distance between the tops of the poles is

The angle of elevation of a cliff at a point $A$ on the ground and a point $B$,$100 \ m$ vertically above $A$,are $\alpha$ and $\beta$ respectively. The height of the cliff is

The shadow of a tower is found to be $60 \ m$ shorter when the sun's altitude changes from $30^{\circ}$ to $60^{\circ}$. The height of the tower from the ground is approximately equal to......$m$

If the angle of elevation of the top of a tower at a distance of $500 \, m$ from its foot is $30^\circ$,then the height of the tower is:

The angle of elevation of a tower from point $P$ due north of it is $30^\circ$ and from a point $Q$ due east of point $P$ is $\theta$. If the distance of the foot of the tower from point $P$ is $3 \ m$ and $PQ = 3 \ m$,then $\theta$ is equal to-