$A$ person standing on the bank of a river observes that the angle subtended by a tree on the opposite bank is $60^\circ$. When he moves $40 \ m$ away from the bank,he finds the angle to be $30^\circ$. The breadth of the river is.....$m$

$P$ 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

$A$ person standing on the top of a building of height $60 \sqrt{3}$ feet observed the top of a tower to lie at an elevation of $45^{\circ}$. That person descended to the bottom of the building and found that the top of the same tower is now at an angle of elevation of $60^{\circ}$. The height of the tower (in feet) is

$A$ tower, of $x$ metres high, has a flagstaff at its top. The tower and the flagstaff subtend equal angles at a point distant $y$ metres from the foot of the tower. Then, the length of the flagstaff (in metres) is:

From the bottom of a pole of height $h,$ the angle of elevation of the top of a tower is $\alpha$ and the pole subtends an angle $\beta$ at the top of the tower. The height of the tower is

$A$ person observes the top of a tower from a point $A$ on the ground. The angle of elevation of the tower from this point is $60^{\circ}$. He moves $60 \ m$ in the direction perpendicular to the line joining $A$ and the base of the tower to reach point $C$. The angle of elevation of the tower from point $C$ is $45^{\circ}$. Then, the height of the tower (in metres) is