Let's break this down step by step.
First, let's define the variables:
- Let \( w \) be the width of the rectangle.
- According to the problem, the length \( l \) can be expressed in terms of the width \( w \) as:
- An expression for the length of the rectangle in terms of the width would be \( l = 2w - 4 \).
Next, we have the formula for the area of a rectangle:
- The formula for the area of a rectangle is \( \text{Area} = l \times w \).
Given that the area is \( 96 , \text{m}^2 \), we can write:
\[ (2w - 4) \cdot w = 96 \]
This expands to:
\[ 2w^2 - 4w = 96 \]
Rearranging this gives us a quadratic equation:
\[ 2w^2 - 4w - 96 = 0 \]
Dividing through by 2 simplifies it:
\[ w^2 - 2w - 48 = 0 \]
Now, we can factor or use the quadratic formula to find \( w \): Factoring gives us:
\[ (w - 8)(w + 6) = 0 \]
Thus, the width \( w \) can be \( 8 \) (the width cannot be negative). Substituting \( w = 8 \) back into the expression for the length:
\[ l = 2(8) - 4 = 16 - 4 = 12 \]
Thus, the dimensions of the rectangle are:
- Using trial and error, if the area is \( 96 , \text{m}^2 \), then the length and width are \( l = 12 , \text{m} \) and \( w = 8 , \text{m} \).
Here is the complete summary with selected options:
- An expression for the length: \( l = 2w - 4 \)
- The formula for the area of a rectangle is: \( \text{Area} = l \times w \)
- Using trial and error, if the area is \( 96 , \text{m}^2 \), then the length and width are: \( l = 12 , \text{m}, w = 8 , \text{m} \)
So your options are:
- \( l = 2w - 4 \)
- \( l = 12 \) & \( w = 8 \)
These two options accurately reflect the problem's requirements.
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