A theodolite is a precision optical instrument for measuring used to measure the speed of sound in a liquid.

A lotus is $20 \mathrm{cm}$ above the water surface in a pond and its stem is partly below the water surface. As the wind blew, the stem is pushed aside so that the lotus touched the water $40 \mathrm{cm}$ away from the original position of the stem. How much of the stem was below the water surface originally? [Write your final answer excluding units]

Draw the diagram for the given information.

A man sees the top of a tower in a mirror which is at a distance of 87.6 m. from the tower. The mirror is on the ground , facing upward . The man is 0.4 m away from the mirror, and the distance of his eye level from the ground is 1.5m. (The foot of man , the mirror and the foot of the tower lie along a straight line.)

A boy standing on a horizontal plane at $B$, watches the top of the flag post ‘$PR$’ at a distance of $100$ $m$ from him at an angle of elevation of $30°$ as shown in the figure. A girl standing at $G$ on the roof of $20 m$ high building ‘$GH$’, finds the angle of elevation of the top of the same flag post to be $45°$. Both the boy and the girl are on the opposite sides of the flag post. Find the distance of the top of the flag post from the girl. (Take $\sqrt{2}=1.41$)

From a point $A$ on a bridge across a river $BC$, the angle of a depression of the banks on opposite sides of river, $B$ and $C$ are $30°$ and $45°$ respectively. If the bridge is at a height of $40m$ as shown in the fig., find the width of the river $BC$.

The angle of elevation of the top of the building from the foot of the tower is $30°$ and the angle of the top of the tower from the foot of the building is $60°$. If the tower is $50 \mathrm{m}$ high, find the height of the building.

A tree is broken by the wind. The top of that tree struck the ground at an angle of $30°$ and at a distance of $30 \mathrm{m}$ from the root. Find the height of the whole tree. $\left(\sqrt{3}=1.73\right)$

As observed from the top of a $100 \mathrm{m}$ high light house from the sea level, the angles of depression of two ships are $30°$ and $45°$. If one ship is exactly behind the other on the same side of the light house, find the distance between the two ships. $[\text{Use} \sqrt{3}=1.732]$