Question
2. 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)
Put responses in the correct input to answer the question. Select a response, navigate to the desired input and insert the response. Responses can be selected and inserted using the space bar, enter key, left mouse button or touchpad. Responses can also be moved by dragging with a mouse.
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
If the area of the rectangle is 96 m^2, what is the length and the width?
(3 points)
Put responses in the correct input to answer the question. Select a response, navigate to the desired input and insert the response. Responses can be selected and inserted using the space bar, enter key, left mouse button or touchpad. Responses can also be moved by dragging with a mouse.
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
GPT-4o mini
To solve this problem, we can break it down into parts.
1. **Express the length in terms of the width**:
The problem states that the length (L) is four meters less than twice the width (W). This can be expressed mathematically as:
\[
L = 2W - 4
\]
2. **Formula for the area of a rectangle**:
The area (A) of a rectangle is calculated as:
\[
A = Length \times Width = L \times W
\]
Substituting for the length in terms of width gives:
\[
A = (2W - 4) \times W
\]
3. **Set the area equal to 96 m²**:
Since the area is given as 96 m², we can set up the equation:
\[
(2W - 4) \times W = 96
\]
This simplifies to:
\[
2W^2 - 4W - 96 = 0
\]
4. **Simplifying the equation**:
We can divide the entire equation by 2:
\[
W^2 - 2W - 48 = 0
\]
5. **Solving the quadratic equation**:
We can factor this equation:
\[
(W - 8)(W + 6) = 0
\]
This gives us two possible solutions:
\[
W = 8 \quad \text{or} \quad W = -6
\]
Since width cannot be negative, we take \(W = 8\) m.
6. **Finding the length**:
Now, substitute \(W\) back to find the length:
\[
L = 2(8) - 4 = 16 - 4 = 12 \text{ m}
\]
So, the final answers are:
- Width: 8 m
- Length: 12 m
### Responses:
- An expression for the length of the rectangle in terms of the width would be **2W - 4**.
- The formula for the area of a rectangle is **L × W**.
- Using trial and error, if the area is 96 m², then the length and width are **Length = 12 m and Width = 8 m**.
1. **Express the length in terms of the width**:
The problem states that the length (L) is four meters less than twice the width (W). This can be expressed mathematically as:
\[
L = 2W - 4
\]
2. **Formula for the area of a rectangle**:
The area (A) of a rectangle is calculated as:
\[
A = Length \times Width = L \times W
\]
Substituting for the length in terms of width gives:
\[
A = (2W - 4) \times W
\]
3. **Set the area equal to 96 m²**:
Since the area is given as 96 m², we can set up the equation:
\[
(2W - 4) \times W = 96
\]
This simplifies to:
\[
2W^2 - 4W - 96 = 0
\]
4. **Simplifying the equation**:
We can divide the entire equation by 2:
\[
W^2 - 2W - 48 = 0
\]
5. **Solving the quadratic equation**:
We can factor this equation:
\[
(W - 8)(W + 6) = 0
\]
This gives us two possible solutions:
\[
W = 8 \quad \text{or} \quad W = -6
\]
Since width cannot be negative, we take \(W = 8\) m.
6. **Finding the length**:
Now, substitute \(W\) back to find the length:
\[
L = 2(8) - 4 = 16 - 4 = 12 \text{ m}
\]
So, the final answers are:
- Width: 8 m
- Length: 12 m
### Responses:
- An expression for the length of the rectangle in terms of the width would be **2W - 4**.
- The formula for the area of a rectangle is **L × W**.
- Using trial and error, if the area is 96 m², then the length and width are **Length = 12 m and Width = 8 m**.