A group of students of class $X$ visited India Gate on an education trip. The teacher and students had interest in history as well. The teacher narrated that India Gate, official name Delhi Memorial, originally called All-India War Memorial, monumental sandstone arch in New Delhi, 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 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?