Heights and Distances

In the given figure, which is the line of sight direction when you are at point $P$ and watching point $O$?

Find the height of the pillar $AB$ shown in the figure.

A spotlight, $\mathrm{P}$, is at a height of $12 \mathrm{m}$ from the surface. The light from $\mathrm{P}$, forms a right circular cone and illuminates a circular region on the surface as shown below.

Find the area of the illuminated circular region on the surface, in terms of $\mathrm{\pi }$. Show your steps along with a rough diagram with all the relevant measures marked.

A vertical tower subtends a right angle at the top of a vertical flag on the ground, the height of the flag being $10 \mathrm{m}$. If the distance between the tower and the flag be $10 \mathrm{m}$, find the height of the tower (in $\mathrm{m}$).

A water tower is located $60$meter from a building (see the figure). From a window in the building, an observer notes that the angle of elevation to the top of the tower is $60°$ and that the angle of depression to the bottom of the tower is $30°$.

How high is the window?

A water tower is located $60$meter from a building (see the figure). From a window in the building, an observer notes that the angle of elevation to the top of the tower is $60°$ and that the angle of depression to the bottom of the tower is $30°$.

How tall is the tower?

For a person standing $324 \mathrm{m}$ from the centre of the base of Eiffel Tower, the angle of elevation to the top of the tower is $45°$. How tall is the Eiffel Tower?

A straight road goes straight up till the base of the building. A man on the top of the building sees a car at $30°$ angle of depression. The car is moving towards the building at a uniform speed. After the car has covered a distance of $30$meters the angle of depression becomes $60°$. If the time taken by the car to reach the building from this point is $10$seconds then find the speed of the car and the height of the building.

There is a vertical column which is $3h \mathrm{meter}$ tall. Find the angle of elevation in degree from a point $\sqrt{3} h \mathrm{meters}$ away from the base of the vertical column.

A tree is $20\sqrt{3}$ meters length. Find the angle of elevation, if its top from a point $20$ meters away from its foot.

If a shadow of a tower standing on a level ground is found to be $60 \mathrm{m}$ longer when sun's elevation is $30°$ than when it is $45°$, then the height of the tower is 

If the shadow of a tower standing on a level ground is found to be $60 \mathrm{m}$ longer when sun's elevation is $30°$ than when it is $45°$, then the height of the tower is

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?

Gadisar Lake is located in the Jaisalmer district of Rajasthan. It was built by the King of Jaisalmer and rebuilt by Gadsi Singh in ${14}^{\mathrm{th}}$ century. The Lake has many Chhatris. One of them is shown below:

Observe the picture. From point $A$ $h \mathrm{m}$ above the water level, the angle of elevation of top of Chhatri (point $B$) is $45°$ and angle of depression to its reflection in water (point $C$)  is $60°$. If the height if Chhatri above water level is (approximately) $10 \mathrm{m}$, then draw a well-labelled figure based on the above information. Find the height $\left(h\right)$ of the point $A$ above water level. (Use $\sqrt{3}=1.73$)

Two poles of heights $25\mathrm{m} \mathrm{and} 35\mathrm{m}$ stand vertically on the ground. The tops of two poles are connected by a wire, which is inclined to the horizontal at an angle of $30°$. Find the length of the wire.

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?

At a point $A, 20 m$ above the level of water in a lake, the angle of elevation of a cloud is $30°$. The angle of depression of the reflection of the cloud in the lake at $A$ is $60°$. Find the distance of the cloud from $A$.

Based on the above information , answer the following questions . (Take $\sqrt{3}=1.73)$

$\left(i\right)$ Find the distance of first position of the pigeon from the eyes of the boy.

$\left(ii\right)$ How much distance the pigeon covers in 8 seconds?

If a tower $30\mathrm{m}$ high, casts a shadow $10\sqrt{3}\mathrm{m}$ long on the ground, then what is the angle of elevation of the sun?

A boy $1.7 \mathrm{m}$ tall is standing on a horizontal ground, $50 \mathrm{m}$ away from a building. The angle of elevation of the top of the building from his eye is $60°$. Calculate the height of the building. $(\mathrm{Take} \sqrt{3}=1.73)$