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?

(3 points)

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

The formula for the area of a rectangle is (BLANK)


using trial and error, if the area is 96 m^2, then the length and width are (BLANK)

l = 12 & w = 8, l = 4 & w = 24, area = 21 + 2w, l = 8 & w = 12, l = 24 & w = 4, l = 4 - 2w, l + 2w - 4, area = l x w, w = 2l - 4, area = 1/2 (l x w)

Answer 1

Let's break down the problem step-by-step.

### Step 1: Set up expressions
1. **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**

2. **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:
1. \( w - 8 = 0 \) → \( w = 8 \) (valid since width must be positive)
2. \( w + 6 = 0 \) → \( w = -6 \) (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
1. The expression for the length of the rectangle in terms of the width would be: **l = 2w - 4**
2. The formula for the area of a rectangle is: **area = l x w**
3. Using trial and error, if the area is 96 m², then the length and width are: **l = 12 & w = 8**.