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Sagot :
Sure! Let's solve the given problem step-by-step.
We start with a wire that is bent into the shape of a rectangle. The given dimensions of the rectangle are:
- Length = 30 cm
- Breadth = 10 cm
### Step 1: Calculate the Perimeter of the Rectangle
The perimeter of a rectangle is given by the formula:
[tex]\[ \text{Perimeter} = 2 \times (\text{Length} + \text{Breadth}) \][/tex]
Substituting the given values:
[tex]\[ \text{Perimeter} = 2 \times (30 \, \text{cm} + 10 \, \text{cm}) \][/tex]
[tex]\[ \text{Perimeter} = 2 \times 40 \, \text{cm} \][/tex]
[tex]\[ \text{Perimeter} = 80 \, \text{cm} \][/tex]
This is the total length of the wire.
### Step 2: Forming the Square
When the wire is rebent into a square, the perimeter of the square is equal to the length of the wire. The perimeter of a square is given by:
[tex]\[ \text{Perimeter} = 4 \times \text{Side} \][/tex]
Given the perimeter of the wire is 80 cm, we set up the equation:
[tex]\[ 80 \, \text{cm} = 4 \times \text{Side} \][/tex]
Solving for the side length:
[tex]\[ \text{Side} = \frac{80 \, \text{cm}}{4} \][/tex]
[tex]\[ \text{Side} = 20 \, \text{cm} \][/tex]
So, the length of each side of the square will be 20 cm.
### Step 3: Area Comparison
#### Area of the Rectangle
The area of a rectangle is given by:
[tex]\[ \text{Area of Rectangle} = \text{Length} \times \text{Breadth} \][/tex]
Substituting the given dimensions:
[tex]\[ \text{Area of Rectangle} = 30 \, \text{cm} \times 10 \, \text{cm} \][/tex]
[tex]\[ \text{Area of Rectangle} = 300 \, \text{cm}^2 \][/tex]
#### Area of the Square
The area of a square is given by:
[tex]\[ \text{Area of Square} = \text{Side}^2 \][/tex]
Using the side length calculated earlier:
[tex]\[ \text{Area of Square} = 20 \, \text{cm} \times 20 \, \text{cm} \][/tex]
[tex]\[ \text{Area of Square} = 400 \, \text{cm}^2 \][/tex]
### Conclusion
Comparing the areas:
- The area of the rectangle is 300 cm².
- The area of the square is 400 cm².
Thus, the square encloses more area than the rectangle.
### Summary of Answers
(a) The length of each side of the square will be 20 cm.
(b) The square encloses more area than the rectangle.
We start with a wire that is bent into the shape of a rectangle. The given dimensions of the rectangle are:
- Length = 30 cm
- Breadth = 10 cm
### Step 1: Calculate the Perimeter of the Rectangle
The perimeter of a rectangle is given by the formula:
[tex]\[ \text{Perimeter} = 2 \times (\text{Length} + \text{Breadth}) \][/tex]
Substituting the given values:
[tex]\[ \text{Perimeter} = 2 \times (30 \, \text{cm} + 10 \, \text{cm}) \][/tex]
[tex]\[ \text{Perimeter} = 2 \times 40 \, \text{cm} \][/tex]
[tex]\[ \text{Perimeter} = 80 \, \text{cm} \][/tex]
This is the total length of the wire.
### Step 2: Forming the Square
When the wire is rebent into a square, the perimeter of the square is equal to the length of the wire. The perimeter of a square is given by:
[tex]\[ \text{Perimeter} = 4 \times \text{Side} \][/tex]
Given the perimeter of the wire is 80 cm, we set up the equation:
[tex]\[ 80 \, \text{cm} = 4 \times \text{Side} \][/tex]
Solving for the side length:
[tex]\[ \text{Side} = \frac{80 \, \text{cm}}{4} \][/tex]
[tex]\[ \text{Side} = 20 \, \text{cm} \][/tex]
So, the length of each side of the square will be 20 cm.
### Step 3: Area Comparison
#### Area of the Rectangle
The area of a rectangle is given by:
[tex]\[ \text{Area of Rectangle} = \text{Length} \times \text{Breadth} \][/tex]
Substituting the given dimensions:
[tex]\[ \text{Area of Rectangle} = 30 \, \text{cm} \times 10 \, \text{cm} \][/tex]
[tex]\[ \text{Area of Rectangle} = 300 \, \text{cm}^2 \][/tex]
#### Area of the Square
The area of a square is given by:
[tex]\[ \text{Area of Square} = \text{Side}^2 \][/tex]
Using the side length calculated earlier:
[tex]\[ \text{Area of Square} = 20 \, \text{cm} \times 20 \, \text{cm} \][/tex]
[tex]\[ \text{Area of Square} = 400 \, \text{cm}^2 \][/tex]
### Conclusion
Comparing the areas:
- The area of the rectangle is 300 cm².
- The area of the square is 400 cm².
Thus, the square encloses more area than the rectangle.
### Summary of Answers
(a) The length of each side of the square will be 20 cm.
(b) The square encloses more area than the rectangle.
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