Let's break the problem down step by step.
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Finding an Expression for the Length in terms of Width: We know that the length (L) of the rectangle is four meters less than twice its width (W). We can express this relationship mathematically: \[ L = 2W - 4 \]
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Formula for the Area of a Rectangle: The area (A) of a rectangle can be calculated using the formula: \[ \text{Area} = L \cdot W \]
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Solving for Length and Width: We know that the area of the rectangle is 96 square meters: \[ L \cdot W = 96 \]
Substituting the expression for L from step 1 into the area formula: \[ (2W - 4) \cdot W = 96 \] Expanding this gives: \[ 2W^2 - 4W = 96 \] Rearranging to form a standard quadratic equation: \[ 2W^2 - 4W - 96 = 0 \] Dividing the entire equation by 2 simplifies it: \[ W^2 - 2W - 48 = 0 \]
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Factoring the Quadratic Equation: We can factor this quadratic: \[ (W - 8)(W + 6) = 0 \] This gives us two potential solutions for W:
- \( W = 8 \) (taking the positive solution, as width cannot be negative)
- \( W = -6 \) (not applicable since width cannot be negative)
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Finding the Length: Now substituting \( W = 8 \) back into the equation for L: \[ L = 2(8) - 4 = 16 - 4 = 12 \]
Thus, the final dimensions of the rectangle are:
- Length: 12 meters
- Width: 8 meters
Summary Answers:
- An expression for the length of the rectangle in terms of the width would be L = 2W - 4.
- The formula for the area of a rectangle is Area = L • W.
- Using trial and error, if the area is 96 m², then the length and width are L = 12 m & W = 8 m.
So, the final correct entries are:
- L = 2W - 4
- Area = L • W
- L = 12 & W = 8