Question

he 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____________

The formula for the area of a rectangle is _____________

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


1. l = 24 & w=4
2. l = 4 - 2w
3. area = 21 + 2w
4.w = 21 - 4
5. area = 1 x w
6. l = 2w - 4
7.l = 4 & w = 24
8. area = 1 over 2 (l x w)
9. l = 12 & w =8
10. l = 8 & w = 12

Answer 1

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:

1. **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:

2. **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:

1. **An expression for the length**: \( l = 2w - 4 \)
2. **The formula for the area of a rectangle is**: \( \text{Area} = l \times w \)
3. **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:

6. \( l = 2w - 4 \)
9. \( l = 12 \) & \( w = 8 \)

These two options accurately reflect the problem's requirements.