Heights and Distances

A ladder 9 m long reaches a point 9 m below the top of a vertical flagstaff. From the foot of the ladder, the elevation of the flagstaff is $60^{\circ}$. What is the height of the flagstaff?

The angles of elevation of the top of a tower standing on a horizontal plane from two points on a line passing through the foot of the tower at distances 49 m and 36 m are $43^{\circ}$ and $47^{\circ}$ respectively.…

A balloon is directly above one end of a bridge. The angle of depression of the other end of the bridge from the balloon is $48^{\circ}$. If the height of the balloon above the bridge is 122 m , then what is the length…

A ladder 6 m long reaches a point 6 m below the top of a vertical flagstaff. From the foot of the ladder, the elevation of the top of the flagstaff is $75^{\circ}$. What is the height of the flagstaff?

Paragraph: A flagstaff 20 m long standing on a pillar 10 m high subtends an angle $\tan ^{-1}(0.5)$ at a point $P$ on the ground. Let $\theta$ be the angle subtended by the pillar at this point $P$. If $x$ is the…

The top of a hill when observed from the top and bottom of a building of height $h$ is at angles of elevation p and q respectively. What is the height of the hill?

The angle of elevation of the top of a tower from a point 20 $m$ away from its base is $45^{\circ}$. What is the height of the tower?

If a flag-staff of 6 m height placed on the top of a tower throws a shadow of $2 \sqrt{3} \mathrm{~m}$ along the ground, then what is the angle that the sun makes with the ground?

From the top of a lighthouse, 100 m high, the angle of depression of a boat is $\tan ^{-1}\left(\frac{5}{12}\right)$. What is the distance between the boat and the lighthouse?

Paragraph: There are two points $P$ and $Q$ due south of a leaning tower, which leans towards north. $P$ is at a distance $x$ and $Q$ is at a distance $y$ from the foot of the tower $(x \gt y)$. The angles of elevation…

A lamp post stands on a horizontal plane. From a point situated at a distance 150 m from its foot, the angle of elevation of the top is $30^{\circ}$. What is the height of the lamp post?

The angles of elevation of the top of a tower from the top and foot of a pole are respectively $30^{\circ}$ and $45^{\circ}$. If $\mathrm{h}_{\mathrm{T}}$ is the height of the tower and $h_p$ is the height of the pole,…