Height and Distance — CDS Elementary Mathematics PYQs

At a point on level ground, the tangent of the angle of elevation of the top of a tower is found to be $\frac{5}{6}$. On walking 70 m towards the tower, the tangent of the angle of elevation of the top of the tower is found to be $\frac{9}{8}$. What is the height of the tower?

Two persons are on diametrically opposite sides of a tower. They measure the angles of elevation of the top of the tower as $30^{\circ }$ and $60^{\circ }$ respectively. If the height of the tower is 100 m, what is the approximate distance between the two persons?

A lamp is kept on a vertical pole. The height of the top of the lamp above the ground is $\frac{5\sqrt{3}}{2}$ m. The perpendicular distances of the bottom of the pole from two adjacent walls meeting perpendicularly are 0.7 m and 2.4 m. What is the distance of the top of the lamp from the corner point of the walls on the ground?

A tower subtends an angle 60° at a point $A$ on the same level as the foot of the tower. $B$ is a point vertically above $A$ and $AB=h$. The angle of depression of the foot of the tower, measured from $B$ is 30°. What is the height of the tower?

Two poles of heights 10 m and 15 m are 25 m apart. What is the height of the point of intersection of the lines joining the tip of each pole to the foot of the other pole?

Two poles are situated 24 m apart and their heights differ by 10 m. What is the distance between their tips?

The angles of elevation of the top of a tower from two points $A$ and $B$ at a distance of $x$ m and $(x+5)$ m from the base of the tower of height 6 m and in the same straight line with it are complementary. What is the value of $x$?

From a point $X$ on a bridge across a river, the angles of depression of two points $P$ and $Q$ on the banks on opposite side of the river are $\alpha$ and $\beta$ respectively. If the point $X$ is at a height $h$ above the surface of the river, what is the width of the river if $\alpha$ and $\beta$ are complementary?

A vertical pole of length 80 m is situated on the horizontal plane. The base of the pole is at $P$. There are two points $A$ and $B$ such that $P$, $A$, $B$ are on the same straight line. Let the angles of elevation of top of the pole from $A$ and $B$ be $\alpha$ and $\beta$ $(\alpha >\beta )$ respectively. If $PA=64$ m and $AB=36$ m, then what is $(\alpha +\beta )$ equal to?

A person on the top of a vertical tower observes a car moving at a uniform speed coming directly towards it. If it takes 6 minutes for the angle of depression to change from 30° to 45°, and further $t$ minutes to reach the tower, which one of the following is correct?