Let's solve each part of the question step-by-step.
### Part 1: Circular Carpet in a Square Room
#### (a) Area of the Room
- The room is square-shaped with each side of length 10 feet.
- The formula for the area of a square is side length squared.
- Therefore, the area of the room is [tex]\( 10 \, \text{ft} \times 10 \, \text{ft} = 100 \, \text{ft}^2 \)[/tex].
#### (b) Area of the Carpet
- The carpet is circular with a diameter of 7 feet.
- The radius of the carpet is half the diameter, so the radius [tex]\( r = 7 \, \text{ft} / 2 = 3.5 \, \text{ft} \)[/tex].
- The formula for the area of a circle is [tex]\(\pi \times \text{radius}^2\)[/tex].
- Given that [tex]\( \pi = \frac{22}{7} \)[/tex],
[tex]\[
\text{Area of the carpet} = \pi \times (3.5 \, \text{ft})^2 = \frac{22}{7} \times 3.5^2 \, \text{ft}^2 = \frac{22}{7} \times 12.25 \, \text{ft}^2 = 38.5 \, \text{ft}^2
\][/tex]
#### (c) Area of the Room Excluding the Carpet
- To find the area of the room excluding the carpet, subtract the area of the carpet from the area of the room.
[tex]\[
\text{Area excluding the carpet} = 100 \, \text{ft}^2 - 38.5 \, \text{ft}^2 = 61.5 \, \text{ft}^2
\][/tex]
### Part 2: Building in a Rectangular Field
#### (a) Total Area of Land
- The building is rectangular with a length of 40 meters and a breadth of 30 meters.
- The area of the building is:
[tex]\[
\text{Building area} = 40 \, \text{m} \times 30 \, \text{m} = 1200 \, \text{m}^2
\][/tex]
- The rectangular field has a length of 200 meters and a breadth of 120 meters.
- The area of the field is:
[tex]\[
\text{Field area} = 200 \, \text{m} \times 120 \, \text{m} = 24000 \, \text{m}^2
\][/tex]
### Final Answer Summary
1. The area of the room: [tex]\( 100 \, \text{ft}^2 \)[/tex]
2. The area of the carpet: [tex]\( 38.5 \, \text{ft}^2 \)[/tex]
3. The area of the room excluding the carpet: [tex]\( 61.5 \, \text{ft}^2 \)[/tex]
4. The area of the building: [tex]\( 1200 \, \text{m}^2 \)[/tex]
5. The area of the rectangular field: [tex]\( 24000 \, \text{m}^2 \)[/tex]
