Let's solve the problem step by step.
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Expression for the length in terms of the width:
Let \( w \) be the width of the rectangle. According to the problem, the length \( l \) is four meters less than twice the width. Thus, we can express the length as: \[ l = 2w - 4 \]
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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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Substituting the expression for length into the area formula:
Since we know the area of the rectangle is \( 96 , \text{m}^2 \), we set up the equation: \[ A = l \times w = 96 \] Substituting the expression for \( l \): \[ (2w - 4) \times w = 96 \]
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Solving the equation:
Expanding this equation gives: \[ 2w^2 - 4w = 96 \] Rearranging it into standard quadratic form: \[ 2w^2 - 4w - 96 = 0 \] Dividing the entire equation by 2 to simplify: \[ w^2 - 2w - 48 = 0 \]
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Factoring the quadratic:
To solve for \( w \), we can factor the quadratic equation: \[ (w - 8)(w + 6) = 0 \] Setting each factor to zero gives: \[ w - 8 = 0 \quad \text{or} \quad w + 6 = 0 \] Thus, we find: \[ w = 8 \quad \text{or} \quad w = -6 \] Since a width cannot be negative, we have: \[ w = 8 , \text{m} \]
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Finding the length:
Now substituting \( w = 8 \) back into the expression for length: \[ l = 2(8) - 4 = 16 - 4 = 12 , \text{m} \]
So, the final dimensions of the rectangle are:
- Width: \( 8 , \text{m} \)
- Length: \( 12 , \text{m} \)
To summarize the answers:
- An expression for the length of the rectangle in terms of the width is: \( l = 2w - 4 \)
- The formula for the area of a rectangle is: \( A = l \times w \)