Let's break down the problem step by step.
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Expression for the Length in terms of the Width: Let the width of the rectangle be \( w \) meters. According to the problem, the length \( l \) is given by: \[ l = 2w - 4 \]
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Formula for the Area of a Rectangle: The area \( A \) of a rectangle is calculated using the formula: \[ A = l \times w \] Given the area is \( 96 , m^2 \), we can substitute the expression for the length into this formula.
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Setting up the equation: Substituting the expression for the length into the area formula gives: \[ 96 = (2w - 4) \times w \] Expanding this equation: \[ 96 = 2w^2 - 4w \] Rearranging it gives: \[ 2w^2 - 4w - 96 = 0 \]
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Simplifying the Equation: Dividing the entire equation by 2: \[ w^2 - 2w - 48 = 0 \]
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Factoring the Quadratic Equation: To factor this equation, we need two numbers that multiply to \(-48\) and add to \(-2\). These numbers are \(-8\) and \(6\): \[ (w - 8)(w + 6) = 0 \]
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Finding the Width: Setting each factor to zero gives: \[ w - 8 = 0 \quad \Rightarrow \quad w = 8 \ w + 6 = 0 \quad \Rightarrow \quad w = -6 \quad (\text{not a valid solution since width cannot be negative}) \] Therefore, the width \( w \) is \( 8 , m \).
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Finding the Length: Now, substituting \( w \) back into the expression for length: \[ l = 2w - 4 = 2(8) - 4 = 16 - 4 = 12 , m \]
Final Answers:
- Width: \( 8 , m \)
- Length: \( 12 , m \)
Summary of responses:
- An expression for the length of the rectangle in terms of the width: \( l = 2w - 4 \)
- The formula for the area of a rectangle: \( A = l \times w \)
- Using trial and error, the dimensions of the rectangle are: Length = \( 12 , m \) and Width = \( 8 , m \)
1 answer