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 areaThe 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 areal=12 & w=8area=12(l×w)l=2w−4area=l×warea=2l+2ww=2l−4l=24 & w=4l=4−2wl=4 & w=24l=8 & w=12
Answers
GPT-4o mini
To solve the problem step by step, we start by defining the variables and relationships provided:
1. **Define the Variables**: Let \( w \) be the width of the rectangle. According to the problem, the length \( l \) is expressed in terms of the width:
\[
l = 2w - 4
\]
2. **Area Formula**: The area \( A \) of a rectangle is given by the formula:
\[
A = l \times w
\]
We know that the area is 96 m²:
\[
96 = l \times w
\]
3. **Substituting for Length**: We can substitute the expression for \( l \) into the area formula:
\[
96 = (2w - 4) \times w
\]
Expanding this gives:
\[
96 = 2w^2 - 4w
\]
Rearranging the equation leads to:
\[
2w^2 - 4w - 96 = 0
\]
4. **Simplifying**: Dividing the entire equation by 2 to make it simpler:
\[
w^2 - 2w - 48 = 0
\]
5. **Factoring**: To factor the quadratic equation, we look for two numbers that multiply to \(-48\) and add to \(-2\). These numbers are \(-8\) and \(6\):
\[
(w - 8)(w + 6) = 0
\]
6. **Finding the Width**: Setting each factor equal to zero gives:
\[
w - 8 = 0 \quad \text{or} \quad w + 6 = 0
\]
This yields \( w = 8 \) (the width cannot be negative).
7. **Finding the Length**: Now substituting \( w = 8 \) back into the equation for \( l \):
\[
l = 2(8) - 4 = 16 - 4 = 12
\]
Summarizing the results:
- The width \( w \) is \( 8 \) meters.
- The length \( l \) is \( 12 \) meters.
Now inserting the responses in the specified input areas:
1. **An expression for the length of the rectangle in terms of the width**: \( 2w - 4 \)
2. **The formula for the area of a rectangle is**: \( l \times w \)
3. **Using trial and error, if the area is 96 m², then the length and width are**: \( l=12 \) & \( w=8 \)
1. **Define the Variables**: Let \( w \) be the width of the rectangle. According to the problem, the length \( l \) is expressed in terms of the width:
\[
l = 2w - 4
\]
2. **Area Formula**: The area \( A \) of a rectangle is given by the formula:
\[
A = l \times w
\]
We know that the area is 96 m²:
\[
96 = l \times w
\]
3. **Substituting for Length**: We can substitute the expression for \( l \) into the area formula:
\[
96 = (2w - 4) \times w
\]
Expanding this gives:
\[
96 = 2w^2 - 4w
\]
Rearranging the equation leads to:
\[
2w^2 - 4w - 96 = 0
\]
4. **Simplifying**: Dividing the entire equation by 2 to make it simpler:
\[
w^2 - 2w - 48 = 0
\]
5. **Factoring**: To factor the quadratic equation, we look for two numbers that multiply to \(-48\) and add to \(-2\). These numbers are \(-8\) and \(6\):
\[
(w - 8)(w + 6) = 0
\]
6. **Finding the Width**: Setting each factor equal to zero gives:
\[
w - 8 = 0 \quad \text{or} \quad w + 6 = 0
\]
This yields \( w = 8 \) (the width cannot be negative).
7. **Finding the Length**: Now substituting \( w = 8 \) back into the equation for \( l \):
\[
l = 2(8) - 4 = 16 - 4 = 12
\]
Summarizing the results:
- The width \( w \) is \( 8 \) meters.
- The length \( l \) is \( 12 \) meters.
Now inserting the responses in the specified input areas:
1. **An expression for the length of the rectangle in terms of the width**: \( 2w - 4 \)
2. **The formula for the area of a rectangle is**: \( l \times w \)
3. **Using trial and error, if the area is 96 m², then the length and width are**: \( l=12 \) & \( w=8 \)