To solve the problem step by step, we'll start with the relationships given.
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Expression for the length of the rectangle in terms of the width: The problem states that the length (l) is four meters less than twice its width (w). This can be expressed as: \[ l = 2w - 4 \] So the response to that part is: 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: \[ A = l \times w \] Therefore, the response here is: area = l x w
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Using the area to find the length and width: We know that the area is 96 m². Using the area formula and substituting the expression for length, we have: \[ 96 = (2w - 4) \times w \] This simplifies to: \[ 96 = 2w^2 - 4w \] Rearranging the equation gives: \[ 2w^2 - 4w - 96 = 0 \] Dividing the entire equation by 2: \[ w^2 - 2w - 48 = 0 \] This can be factored as: \[ (w - 8)(w + 6) = 0 \] So the two possible values for w are: \[ w = 8 \quad \text{(Positive width)} \ w = -6 \quad \text{(Not valid as width cannot be negative)} \] Thus, \( w = 8 \) m. Now, using the expression for length: \[ l = 2(8) - 4 = 16 - 4 = 12 , m \]
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Final answers: Width \( w = 8 , m \) and Length \( l = 12 , m \).
Thus, the final response for the width and length is: l = 12 & w = 8.
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