Let's break down the problem step-by-step.
Step 1: Set up expressions
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Length in terms of Width: According to the first statement, "The length of a rectangle is four meters less than twice its width."
We can express the length \( l \) in terms of the width \( w \) as: \[ l = 2w - 4 \]So, the expression for the length of the rectangle in terms of the width would be:
l = 2w - 4 -
Formula for the Area of a Rectangle: The area \( A \) of a rectangle is calculated using the formula: \[ A = l \times w \]
So, the formula for the area of a rectangle is:
area = l x w
Step 2: Solve for dimensions
We are given that the area of the rectangle is 96 m². We can substitute the expression for the length into the area formula: \[ A = (2w - 4) \times w \] Setting the area equal to 96 m² gives us: \[ (2w - 4) \times w = 96 \] Expanding this: \[ 2w^2 - 4w = 96 \] Rearranging to set it to zero: \[ 2w^2 - 4w - 96 = 0 \] Dividing the entire equation by 2: \[ w^2 - 2w - 48 = 0 \]
Step 3: Factor the quadratic equation
To solve this quadratic equation, we can factor it: \[ (w - 8)(w + 6) = 0 \] Setting each factor to zero gives solutions:
- \( w - 8 = 0 \) → \( w = 8 \) (valid since width must be positive)
- \( w + 6 = 0 \) → \( w = -6 \) (not valid since width cannot be negative)
Step 4: Find the Length
Now that we have \( w = 8 \): Using the expression we derived for \( l \): \[ l = 2w - 4 = 2(8) - 4 = 16 - 4 = 12 \]
Final Dimensions
Thus, the dimensions of the rectangle are:
- Width (w) = 8 m
- Length (l) = 12 m
Answer Summary
- The 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 x w
- Using trial and error, if the area is 96 m², then the length and width are: l = 12 & w = 8.
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