Question

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

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 please

Answer 1

Let's break down the problem step by step.

1. **Expression for the Length in terms of the Width**:
Let the width of the rectangle be \( w \) meters. According to the problem, the length \( l \) is given by:
\[
l = 2w - 4
\]

2. **Formula for the Area of a Rectangle**:
The area \( A \) of a rectangle is calculated using the formula:
\[
A = l \times w
\]
Given the area is \( 96 \, m^2 \), we can substitute the expression for the length into this formula.

3. **Setting up the equation**:
Substituting the expression for the length into the area formula gives:
\[
96 = (2w - 4) \times w
\]
Expanding this equation:
\[
96 = 2w^2 - 4w
\]
Rearranging it gives:
\[
2w^2 - 4w - 96 = 0
\]

4. **Simplifying the Equation**:
Dividing the entire equation by 2:
\[
w^2 - 2w - 48 = 0
\]

5. **Factoring the Quadratic Equation**:
To factor this equation, we need two numbers that multiply to \(-48\) and add to \(-2\). These numbers are \(-8\) and \(6\):
\[
(w - 8)(w + 6) = 0
\]

6. **Finding the Width**:
Setting each factor to zero gives:
\[
w - 8 = 0 \quad \Rightarrow \quad w = 8 \\
w + 6 = 0 \quad \Rightarrow \quad w = -6 \quad (\text{not a valid solution since width cannot be negative})
\]
Therefore, the width \( w \) is \( 8 \, m \).

7. **Finding the Length**:
Now, substituting \( w \) back into the expression for length:
\[
l = 2w - 4 = 2(8) - 4 = 16 - 4 = 12 \, m
\]

**Final Answers**:
- **Width**: \( 8 \, m \)
- **Length**: \( 12 \, m \)

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