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) 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%C2%A0Response area The formula for the area of a rectangle is%C2%A0Response area%C2%A0 Using trial and error, if the area is 96 m^2, then the length and width are
Answers
GPT-4o mini
Let's break down the problem step-by-step.
1. **Define Variables**:
Let \( w \) be the width of the rectangle. Then the length \( l \) can be expressed in terms of the width:
\[
l = 2w - 4
\]
2. **Area Formula**:
The formula for the area of a rectangle is:
\[
\text{Area} = \text{length} \times \text{width}
\]
Therefore, we can write:
\[
l \times w = 96
\]
3. **Substituting the Expression for Length**:
Substitute \( l \) with \( 2w - 4 \):
\[
(2w - 4) \times w = 96
\]
Expanding this gives:
\[
2w^2 - 4w = 96
\]
Rearranging the equation:
\[
2w^2 - 4w - 96 = 0
\]
Dividing the entire equation by 2 simplifies it to:
\[
w^2 - 2w - 48 = 0
\]
4. **Factoring the Quadratic**:
We will look for two numbers that multiply to \(-48\) and add up to \(-2\). These numbers are \(-8\) and \(6\). Therefore, we can factor the quadratic as follows:
\[
(w - 8)(w + 6) = 0
\]
5. **Finding Solutions**:
This gives us:
\[
w - 8 = 0 \quad \Rightarrow \quad w = 8
\]
\[
w + 6 = 0 \quad \Rightarrow \quad w = -6 \quad \text{(not valid since width cannot be negative)}
\]
6. **Calculating Length**:
Now we found \( w = 8 \) m. Using the expression for length:
\[
l = 2w - 4 = 2(8) - 4 = 16 - 4 = 12 \, m
\]
**Final Answers**:
- Width: 8 m
- Length: 12 m
So the responses should be:
- 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 \(l \times w\).
- Using trial and error, if the area is 96 m², then the length and width are 12 m and 8 m, respectively.
1. **Define Variables**:
Let \( w \) be the width of the rectangle. Then the length \( l \) can be expressed in terms of the width:
\[
l = 2w - 4
\]
2. **Area Formula**:
The formula for the area of a rectangle is:
\[
\text{Area} = \text{length} \times \text{width}
\]
Therefore, we can write:
\[
l \times w = 96
\]
3. **Substituting the Expression for Length**:
Substitute \( l \) with \( 2w - 4 \):
\[
(2w - 4) \times w = 96
\]
Expanding this gives:
\[
2w^2 - 4w = 96
\]
Rearranging the equation:
\[
2w^2 - 4w - 96 = 0
\]
Dividing the entire equation by 2 simplifies it to:
\[
w^2 - 2w - 48 = 0
\]
4. **Factoring the Quadratic**:
We will look for two numbers that multiply to \(-48\) and add up to \(-2\). These numbers are \(-8\) and \(6\). Therefore, we can factor the quadratic as follows:
\[
(w - 8)(w + 6) = 0
\]
5. **Finding Solutions**:
This gives us:
\[
w - 8 = 0 \quad \Rightarrow \quad w = 8
\]
\[
w + 6 = 0 \quad \Rightarrow \quad w = -6 \quad \text{(not valid since width cannot be negative)}
\]
6. **Calculating Length**:
Now we found \( w = 8 \) m. Using the expression for length:
\[
l = 2w - 4 = 2(8) - 4 = 16 - 4 = 12 \, m
\]
**Final Answers**:
- Width: 8 m
- Length: 12 m
So the responses should be:
- 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 \(l \times w\).
- Using trial and error, if the area is 96 m², then the length and width are 12 m and 8 m, respectively.
responses: area = 21+ 20
area = 1xw
20=21-4
::
area = (1xw)
1 = 12 &w=8
1 = 4 & w = 24
1 = 8 & w = 12
= 24 & w = 4
14-20
1 = 20-4
area = 1xw
20=21-4
::
area = (1xw)
1 = 12 &w=8
1 = 4 & w = 24
1 = 8 & w = 12
= 24 & w = 4
14-20
1 = 20-4
GPT-4o mini
It looks like you're trying to manipulate equations for the area of the rectangle and the relationship between length and width based on your earlier question. However, the format appears a bit jumbled. Let's clarify and summarize the proper responses based on the given problem.
### Responses Based on the Problem:
1. **Expression for the Length of the Rectangle in Terms of Width**:
- \( \text{length} = 2w - 4 \)
2. **Formula for the Area of a Rectangle**:
- \( \text{Area} = \text{length} \times \text{width} \)
3. **Using Known Area**:
- Given that the area is \( 96 \, m^2 \) and substituting into the area formula:
\[
(2w - 4) \times w = 96
\]
4. **Trying Specific Values (Trial and Error)**:
After calculations, we found two valid width-length pairs:
- If \( w = 8 \) m, then \( l = 12 \) m; this gives \( area = 12 \times 8 = 96 \, m^2 \).
- If we trial with invalid combinations such as \( w = 4 \, m; l = 16 \, m \), which provides \( area = 64 \, m^2 \), it doesn't satisfy.
- Trying \( w = 12 \, m; l = 20 \, m \), gives \( area = 240 \, m^2\), which is also invalid.
### Finalized Responses:
- **Length Expression**: \(2w - 4\)
- **Area Formula**: \( l \times w \)
- **Valid Pair**: Length = 12 m; Width = 8 m (since \( 12 \cdot 8 = 96 \, m^2\))
If you're using specific numerical responses in a particular input format, please clarify further so I can help organize them better for submission.
### Responses Based on the Problem:
1. **Expression for the Length of the Rectangle in Terms of Width**:
- \( \text{length} = 2w - 4 \)
2. **Formula for the Area of a Rectangle**:
- \( \text{Area} = \text{length} \times \text{width} \)
3. **Using Known Area**:
- Given that the area is \( 96 \, m^2 \) and substituting into the area formula:
\[
(2w - 4) \times w = 96
\]
4. **Trying Specific Values (Trial and Error)**:
After calculations, we found two valid width-length pairs:
- If \( w = 8 \) m, then \( l = 12 \) m; this gives \( area = 12 \times 8 = 96 \, m^2 \).
- If we trial with invalid combinations such as \( w = 4 \, m; l = 16 \, m \), which provides \( area = 64 \, m^2 \), it doesn't satisfy.
- Trying \( w = 12 \, m; l = 20 \, m \), gives \( area = 240 \, m^2\), which is also invalid.
### Finalized Responses:
- **Length Expression**: \(2w - 4\)
- **Area Formula**: \( l \times w \)
- **Valid Pair**: Length = 12 m; Width = 8 m (since \( 12 \cdot 8 = 96 \, m^2\))
If you're using specific numerical responses in a particular input format, please clarify further so I can help organize them better for submission.