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Sagot :
Let's break this problem down into sections and solve each part step-by-step.
### Part (a): Calculating the perimeter of the rectangle
The formula for the perimeter \(P\) of a rectangle is:
[tex]\[ P = 2 \times (\text{length} + \text{breadth}) \][/tex]
Given the length \( l = 4x - 3y \) and the breadth \( b = 2x + y \), we substitute these into the formula:
[tex]\[ P = 2 \times ((4x - 3y) + (2x + y)) \][/tex]
Now, combine like terms inside the parentheses:
[tex]\[ P = 2 \times (4x + 2x - 3y + y) \][/tex]
[tex]\[ P = 2 \times (6x - 2y) \][/tex]
Now, distribute the 2:
[tex]\[ P = 12x - 4y \][/tex]
So, the perimeter of the rectangle in terms of \( x \) and \( y \) is:
[tex]\[ P = 12x - 4y \][/tex]
### Part (b): Calculating the area of the rectangle
The formula for the area \(A\) of a rectangle is:
[tex]\[ A = \text{length} \times \text{breadth} \][/tex]
Substituting the given expressions for length and breadth:
[tex]\[ A = (4x - 3y) \times (2x + y) \][/tex]
We need to expand this product:
[tex]\[ A = (4x - 3y)(2x + y) \][/tex]
Use the distributive property (also known as the FOIL method for binomials):
[tex]\[ A = 4x \cdot 2x + 4x \cdot y - 3y \cdot 2x - 3y \cdot y \][/tex]
[tex]\[ A = 8x^2 + 4xy - 6xy - 3y^2 \][/tex]
Now, combine like terms:
[tex]\[ A = 8x^2 - 2xy - 3y^2 \][/tex]
So, the area of the rectangle in terms of \( x \) and \( y \) is:
[tex]\[ A = 8x^2 - 2xy - 3y^2 \][/tex]
### Part (c): Calculating the actual perimeter and area for \( x = 3 \, \text{cm} \) and \( y = 2 \, \text{cm} \)
First, substitute \( x = 3 \) and \( y = 2 \) into the expressions for length and breadth:
[tex]\[ \text{length} = 4x - 3y = 4(3) - 3(2) = 12 - 6 = 6 \, \text{cm} \][/tex]
[tex]\[ \text{breadth} = 2x + y = 2(3) + 2 = 6 + 2 = 8 \, \text{cm} \][/tex]
#### Actual perimeter
Now, use these values to find the actual perimeter:
[tex]\[ P = 2 \times (\text{length} + \text{breadth}) \][/tex]
[tex]\[ P = 2 \times (6 \, \text{cm} + 8 \, \text{cm}) \][/tex]
[tex]\[ P = 2 \times 14 \, \text{cm} \][/tex]
[tex]\[ P = 28 \, \text{cm} \][/tex]
#### Actual area
Next, calculate the actual area using the actual length and breadth:
[tex]\[ A = \text{length} \times \text{breadth} \][/tex]
[tex]\[ A = 6 \, \text{cm} \times 8 \, \text{cm} \][/tex]
[tex]\[ A = 48 \, \text{cm}^2 \][/tex]
### Summary
(a) The perimeter of the rectangle in terms of \( x \) and \( y \) is:
[tex]\[ P = 12x - 4y \][/tex]
(b) The area of the rectangle in terms of \( x \) and \( y \) is:
[tex]\[ A = 8x^2 - 2xy - 3y^2 \][/tex]
(c) For \( x = 3 \, \text{cm} \) and \( y = 2 \, \text{cm} \):
- The actual perimeter is \( 28 \, \text{cm} \)
- The actual area is [tex]\( 48 \, \text{cm}^2 \)[/tex]
### Part (a): Calculating the perimeter of the rectangle
The formula for the perimeter \(P\) of a rectangle is:
[tex]\[ P = 2 \times (\text{length} + \text{breadth}) \][/tex]
Given the length \( l = 4x - 3y \) and the breadth \( b = 2x + y \), we substitute these into the formula:
[tex]\[ P = 2 \times ((4x - 3y) + (2x + y)) \][/tex]
Now, combine like terms inside the parentheses:
[tex]\[ P = 2 \times (4x + 2x - 3y + y) \][/tex]
[tex]\[ P = 2 \times (6x - 2y) \][/tex]
Now, distribute the 2:
[tex]\[ P = 12x - 4y \][/tex]
So, the perimeter of the rectangle in terms of \( x \) and \( y \) is:
[tex]\[ P = 12x - 4y \][/tex]
### Part (b): Calculating the area of the rectangle
The formula for the area \(A\) of a rectangle is:
[tex]\[ A = \text{length} \times \text{breadth} \][/tex]
Substituting the given expressions for length and breadth:
[tex]\[ A = (4x - 3y) \times (2x + y) \][/tex]
We need to expand this product:
[tex]\[ A = (4x - 3y)(2x + y) \][/tex]
Use the distributive property (also known as the FOIL method for binomials):
[tex]\[ A = 4x \cdot 2x + 4x \cdot y - 3y \cdot 2x - 3y \cdot y \][/tex]
[tex]\[ A = 8x^2 + 4xy - 6xy - 3y^2 \][/tex]
Now, combine like terms:
[tex]\[ A = 8x^2 - 2xy - 3y^2 \][/tex]
So, the area of the rectangle in terms of \( x \) and \( y \) is:
[tex]\[ A = 8x^2 - 2xy - 3y^2 \][/tex]
### Part (c): Calculating the actual perimeter and area for \( x = 3 \, \text{cm} \) and \( y = 2 \, \text{cm} \)
First, substitute \( x = 3 \) and \( y = 2 \) into the expressions for length and breadth:
[tex]\[ \text{length} = 4x - 3y = 4(3) - 3(2) = 12 - 6 = 6 \, \text{cm} \][/tex]
[tex]\[ \text{breadth} = 2x + y = 2(3) + 2 = 6 + 2 = 8 \, \text{cm} \][/tex]
#### Actual perimeter
Now, use these values to find the actual perimeter:
[tex]\[ P = 2 \times (\text{length} + \text{breadth}) \][/tex]
[tex]\[ P = 2 \times (6 \, \text{cm} + 8 \, \text{cm}) \][/tex]
[tex]\[ P = 2 \times 14 \, \text{cm} \][/tex]
[tex]\[ P = 28 \, \text{cm} \][/tex]
#### Actual area
Next, calculate the actual area using the actual length and breadth:
[tex]\[ A = \text{length} \times \text{breadth} \][/tex]
[tex]\[ A = 6 \, \text{cm} \times 8 \, \text{cm} \][/tex]
[tex]\[ A = 48 \, \text{cm}^2 \][/tex]
### Summary
(a) The perimeter of the rectangle in terms of \( x \) and \( y \) is:
[tex]\[ P = 12x - 4y \][/tex]
(b) The area of the rectangle in terms of \( x \) and \( y \) is:
[tex]\[ A = 8x^2 - 2xy - 3y^2 \][/tex]
(c) For \( x = 3 \, \text{cm} \) and \( y = 2 \, \text{cm} \):
- The actual perimeter is \( 28 \, \text{cm} \)
- The actual area is [tex]\( 48 \, \text{cm}^2 \)[/tex]
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