The length of a rectangle is four meters less than twice its width.

Question

If the area of the rectangle is 96 m^2, what is the length and the width?

An expression for the length of the rectangle in terms of the width would be Response area

The formula for the area of a rectangle is Response area

Using trial and error, if the area is 96 m^2, then the length and width are Response area

answers pleaseeeee

Answer 1

To solve the problem, we'll start by establishing the relationships between the length and the width of the rectangle based on the given information.

1. Expression for the length in terms of the width: Let \( w \) be the width. According to the problem, the length \( l \) is four meters less than twice its width. So, we can express this as: \[ l = 2w - 4 \]

2. Formula for the area of a rectangle: The area \( A \) of a rectangle is given by: \[ A = l \times w \] Since we know the area is \( 96 , m^2 \), we write: \[ 96 = l \times w \]

3. Substituting the expression for length into the area formula: We substitute the expression for \( l \) from step 1 into the area formula: \[ 96 = (2w - 4) \times w \] This simplifies to: \[ 96 = 2w^2 - 4w \] Rearranging gives us a standard quadratic equation: \[ 2w^2 - 4w - 96 = 0 \] Dividing the entire equation by 2 to simplify: \[ w^2 - 2w - 48 = 0 \]

4. Solving the quadratic equation: We can factor this equation: \[ (w - 8)(w + 6) = 0 \] Therefore, the solutions for \( w \) are: \[ w = 8 \quad \text{or} \quad w = -6 \] Since width cannot be negative, we have: \[ w = 8 , m \]

5. Finding the length: Now we can find the length using the expression for \( l \): \[ l = 2w - 4 = 2(8) - 4 = 16 - 4 = 12 , m \]

So the final answers are:

• Width: \( 8 , m \)
• Length: \( 12 , m \)

Final Responses:

1. An expression for the length in terms of the width: \( l = 2w - 4 \)
2. The formula for the area of a rectangle: \( A = l \times w \)
3. The length and width are: Length = \( 12 , m \) and Width = \( 8 , m \)