Let's break down the problem step-by-step based on the information provided.
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Expression for the length in terms of the width: The problem states that the length (\(l\)) is four meters less than twice its width (\(w\)). Therefore: \[ l = 2w - 4 \]
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Formula for the area of a rectangle: The area (\(A\)) of a rectangle is given by the formula: \[ A = l \times w \]
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Using the area to set up the equation: We know the area of the rectangle is 96 m², thus: \[ l \times w = 96 \] Substituting the expression for \(l\) from step 1: \[ (2w - 4) \times w = 96 \]
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Expanding and rearranging: Expanding the equation gives: \[ 2w^2 - 4w = 96 \] Rearranging it: \[ 2w^2 - 4w - 96 = 0 \] Dividing the entire equation by 2 to simplify: \[ w^2 - 2w - 48 = 0 \]
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Factoring the quadratic equation: To factor \(w^2 - 2w - 48 = 0\), we look for two numbers that multiply to -48 and add to -2. The numbers are -8 and 6. Therefore: \[ (w - 8)(w + 6) = 0 \]
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Solving for \(w\): Setting each factor equal to zero gives: \[ w - 8 = 0 \quad \Rightarrow \quad w = 8 \] \[ w + 6 = 0 \quad \Rightarrow \quad w = -6 \quad (\text{not valid since width can't be negative}) \]
Therefore, \(w = 8\) meters.
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Finding the corresponding length: Now, substitute \(w = 8\) back into the expression for \(l\): \[ l = 2(8) - 4 = 16 - 4 = 12 \text{ meters} \]
In conclusion:
- The width is \(8\) meters.
- The length is \(12\) meters.
You can now fill out the responses accordingly:
- 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, if the area is 96 m², then the length and width are: \(l = 12 \text{ m} & w = 8 \text{ m}\)