Answer :
Certainly! For each part of the problem, let's analyze the changes step-by-step:
### Initial Area Calculation
First, let's denote the initial length of the rectangle as [tex]\( l \)[/tex] and the initial breadth as [tex]\( b \)[/tex]. The area [tex]\( A \)[/tex] of a rectangle is given by the formula:
[tex]\[ A = l \times b \][/tex]
### Part (i) - Doubling Dimensions Separately
1. Length is doubled, breadth remains the same:
- New length = [tex]\( 2l \)[/tex]
- Breadth = [tex]\( b \)[/tex]
- New area [tex]\( A' \)[/tex] is calculated as:
[tex]\[ A' = (2l) \times b = 2 \times (l \times b) = 2A \][/tex]
- Hence, the area is doubled.
2. Breadth is doubled, length remains the same:
- Length = [tex]\( l \)[/tex]
- New breadth = [tex]\( 2b \)[/tex]
- New area [tex]\( A'' \)[/tex] is calculated as:
[tex]\[ A'' = l \times (2b) = 2 \times (l \times b) = 2A \][/tex]
- Hence, the area is again doubled.
### Part (ii) - Doubling Both Dimensions
- Both length and breadth are doubled:
- New length = [tex]\( 2l \)[/tex]
- New breadth = [tex]\( 2b \)[/tex]
- New area [tex]\( A''' \)[/tex] is calculated as:
[tex]\[ A''' = (2l) \times (2b) = 4 \times (l \times b) = 4A \][/tex]
- Hence, the area is quadrupled.
### Summary
- When the length is doubled while the breadth remains the same, the area is doubled.
- When the breadth is doubled while the length remains the same, the area is also doubled.
- When both the length and breadth are doubled, the area becomes four times the original area.
### Initial Area Calculation
First, let's denote the initial length of the rectangle as [tex]\( l \)[/tex] and the initial breadth as [tex]\( b \)[/tex]. The area [tex]\( A \)[/tex] of a rectangle is given by the formula:
[tex]\[ A = l \times b \][/tex]
### Part (i) - Doubling Dimensions Separately
1. Length is doubled, breadth remains the same:
- New length = [tex]\( 2l \)[/tex]
- Breadth = [tex]\( b \)[/tex]
- New area [tex]\( A' \)[/tex] is calculated as:
[tex]\[ A' = (2l) \times b = 2 \times (l \times b) = 2A \][/tex]
- Hence, the area is doubled.
2. Breadth is doubled, length remains the same:
- Length = [tex]\( l \)[/tex]
- New breadth = [tex]\( 2b \)[/tex]
- New area [tex]\( A'' \)[/tex] is calculated as:
[tex]\[ A'' = l \times (2b) = 2 \times (l \times b) = 2A \][/tex]
- Hence, the area is again doubled.
### Part (ii) - Doubling Both Dimensions
- Both length and breadth are doubled:
- New length = [tex]\( 2l \)[/tex]
- New breadth = [tex]\( 2b \)[/tex]
- New area [tex]\( A''' \)[/tex] is calculated as:
[tex]\[ A''' = (2l) \times (2b) = 4 \times (l \times b) = 4A \][/tex]
- Hence, the area is quadrupled.
### Summary
- When the length is doubled while the breadth remains the same, the area is doubled.
- When the breadth is doubled while the length remains the same, the area is also doubled.
- When both the length and breadth are doubled, the area becomes four times the original area.
