Let's start by defining the variables for the rectangle:
Let \( w \) be the width of the rectangle. According to the problem, the length \( l \) can be expressed in terms of the width as follows:
Expression for the length in terms of width: \[ l = 2w - 4 \]
The formula for the area \( A \) of a rectangle is: \[ A = l \times w \]
Given that the area is \( 96 , \text{m}^2 \), we can substitute the expression for the length into the formula for the area:
\[ 96 = (2w - 4) \times w \]
Expanding this, we have: \[ 96 = 2w^2 - 4w \]
Rearranging the equation gives: \[ 2w^2 - 4w - 96 = 0 \]
To simplify, we can divide the entire equation by 2: \[ w^2 - 2w - 48 = 0 \]
Next, we can factor this quadratic equation. We need to find two numbers that multiply to \(-48\) and add to \(-2\). The numbers \(-8\) and \(6\) work here:
So, we can factor the equation as: \[ (w - 8)(w + 6) = 0 \]
Setting each factor to zero gives us:
- \( w - 8 = 0 \) which gives \( w = 8 \)
- \( w + 6 = 0 \) which gives \( w = -6 \) (not a valid solution since width cannot be negative)
Thus, the width is: \[ w = 8 , \text{m} \]
Now we can find the length using the expression for length in terms of width: \[ l = 2(8) - 4 = 16 - 4 = 12 , \text{m} \]
So, the dimensions of the rectangle are:
Length: \( 12 , \text{m} \)
Width: \( 8 , \text{m} \)
To summarize:
- An expression for the length of the rectangle in terms of width is: \[ l = 2w - 4 \]
- The formula for the area of a rectangle is: \[ A = l \times w \]
- Using trial and error, the values we found for the area of \( 96 , \text{m}^2 \) yields dimensions: \[ \text{Length} = 12 , \text{m}, \quad \text{Width} = 8 , \text{m} \]
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