A group of students of class $\mathrm{X}$ visited India Gate on an education trip. The teacher and students had an interest in history as well. The teacher narrated that India Gate, officially named Delhi Memorial, originally called All-India War Memorial, the monumental sandstone arch in New Delhi, was dedicated to the troops of British India who died in wars fought between $1914$ and $1919$.

The teacher also said that India Gate, which is located at the eastern end of the Rajpath (formerly called the Kingsway), is about $138 \mathrm{feet} (42 \mathrm{metres})$ in height. What is the angle of elevation (in degrees) if they are standing at a distance of $42 \mathrm{m}$ away from the monument?

A ladder $10 m$ long is leaning against a vertical wall. It makes an angle of $60°$ with the ground. How far is the foot of the ladder from the wall?

A sphere with centre $O$ sits atop a pole as shown in the figure. An observer on the ground is at a distance $50 m$ from the foot of the pole. She notes that the angles of elevation from the observer to points $P$ and $Q$ on the sphere are $30°$ and $60°$ , respectively. Then, the radius of the sphere in meters is

Two poles of the height $12 m$ and $17 m$ stand vertically upright on a plain ground. If the distance between their feet is $12 m$, find the distance between their tops.

X is $5 m$ tall and he notices that he casts a shadow that is $3 m$ long. He then measures that the shadow cast by his school building is $30 m$ long. How tall is the building?