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 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
Answers
GPT-4o mini
Let's break down the problem step by step.
1. **Define Variables:**
Let the width of the rectangle be \( w \).
The length \( l \) can be expressed in terms of the width:
\[
l = 2w - 4
\]
2. **Area of a Rectangle:**
The formula for the area of a rectangle is:
\[
\text{Area} = \text{length} \times \text{width}
\]
Therefore, substituting the expressions for length and width:
\[
96 = (2w - 4) \times w
\]
3. **Set Up the Equation:**
Expanding the equation, we get:
\[
96 = 2w^2 - 4w
\]
Rearranging it gives:
\[
2w^2 - 4w - 96 = 0
\]
4. **Simplify the Quadratic Equation:**
Dividing the entire equation by 2:
\[
w^2 - 2w - 48 = 0
\]
5. **Factor the Quadratic:**
Looking for two numbers that multiply to -48 and add to -2, we can factor:
\[
(w - 8)(w + 6) = 0
\]
This gives us the solutions:
\[
w - 8 = 0 \quad \Rightarrow \quad w = 8
\]
\[
w + 6 = 0 \quad \Rightarrow \quad w = -6 \quad (\text{not valid since width cannot be negative})
\]
6. **Calculate Length:**
Now substituting \( w = 8 \) back into the expression for length:
\[
l = 2(8) - 4 = 16 - 4 = 12
\]
7. **Final Answers:**
- The width \( w \) is 8 meters.
- The length \( l \) is 12 meters.
Putting the responses in the correct input:
- An expression for the length of the rectangle in terms of the width would be: **\( l = 2w - 4 \)**
- The formula for the area of a rectangle is: **\( \text{Area} = l \times w \)**
- Using trial and error, if the area is 96 m², then the length and width are: **Length = 12 m, Width = 8 m**.
1. **Define Variables:**
Let the width of the rectangle be \( w \).
The length \( l \) can be expressed in terms of the width:
\[
l = 2w - 4
\]
2. **Area of a Rectangle:**
The formula for the area of a rectangle is:
\[
\text{Area} = \text{length} \times \text{width}
\]
Therefore, substituting the expressions for length and width:
\[
96 = (2w - 4) \times w
\]
3. **Set Up the Equation:**
Expanding the equation, we get:
\[
96 = 2w^2 - 4w
\]
Rearranging it gives:
\[
2w^2 - 4w - 96 = 0
\]
4. **Simplify the Quadratic Equation:**
Dividing the entire equation by 2:
\[
w^2 - 2w - 48 = 0
\]
5. **Factor the Quadratic:**
Looking for two numbers that multiply to -48 and add to -2, we can factor:
\[
(w - 8)(w + 6) = 0
\]
This gives us the solutions:
\[
w - 8 = 0 \quad \Rightarrow \quad w = 8
\]
\[
w + 6 = 0 \quad \Rightarrow \quad w = -6 \quad (\text{not valid since width cannot be negative})
\]
6. **Calculate Length:**
Now substituting \( w = 8 \) back into the expression for length:
\[
l = 2(8) - 4 = 16 - 4 = 12
\]
7. **Final Answers:**
- The width \( w \) is 8 meters.
- The length \( l \) is 12 meters.
Putting the responses in the correct input:
- An expression for the length of the rectangle in terms of the width would be: **\( l = 2w - 4 \)**
- The formula for the area of a rectangle is: **\( \text{Area} = l \times w \)**
- Using trial and error, if the area is 96 m², then the length and width are: **Length = 12 m, Width = 8 m**.