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? An expression for the length of rectangle in terms of width would be ____. The formula for the area of a rectangle is _____. Using trial and error, if the area is 96^2, then the length and width are ___.
Answers
GPT-4o mini
Let's start by defining the variables for the rectangle:
Let \( w \) be the width of the rectangle. According to the problem, the length \( l \) can be expressed in terms of the width as follows:
**Expression for the length in terms of width**:
\[
l = 2w - 4
\]
The formula for the area \( A \) of a rectangle is:
\[
A = l \times w
\]
Given that the area is \( 96 \, \text{m}^2 \), we can substitute the expression for the length into the formula for the area:
\[
96 = (2w - 4) \times w
\]
Expanding this, we have:
\[
96 = 2w^2 - 4w
\]
Rearranging the equation gives:
\[
2w^2 - 4w - 96 = 0
\]
To simplify, we can divide the entire equation by 2:
\[
w^2 - 2w - 48 = 0
\]
Next, we can factor this quadratic equation. We need to find two numbers that multiply to \(-48\) and add to \(-2\). The numbers \(-8\) and \(6\) work here:
So, we can factor the equation as:
\[
(w - 8)(w + 6) = 0
\]
Setting each factor to zero gives us:
1. \( w - 8 = 0 \) which gives \( w = 8 \)
2. \( w + 6 = 0 \) which gives \( w = -6 \) (not a valid solution since width cannot be negative)
Thus, the width is:
\[
w = 8 \, \text{m}
\]
Now we can find the length using the expression for length in terms of width:
\[
l = 2(8) - 4 = 16 - 4 = 12 \, \text{m}
\]
So, the dimensions of the rectangle are:
**Length**: \( 12 \, \text{m} \)
**Width**: \( 8 \, \text{m} \)
To summarize:
- An expression for the length of the rectangle in terms of width is:
\[
l = 2w - 4
\]
- The formula for the area of a rectangle is:
\[
A = l \times w
\]
- Using trial and error, the values we found for the area of \( 96 \, \text{m}^2 \) yields dimensions:
\[
\text{Length} = 12 \, \text{m}, \quad \text{Width} = 8 \, \text{m}
\]
Let \( w \) be the width of the rectangle. According to the problem, the length \( l \) can be expressed in terms of the width as follows:
**Expression for the length in terms of width**:
\[
l = 2w - 4
\]
The formula for the area \( A \) of a rectangle is:
\[
A = l \times w
\]
Given that the area is \( 96 \, \text{m}^2 \), we can substitute the expression for the length into the formula for the area:
\[
96 = (2w - 4) \times w
\]
Expanding this, we have:
\[
96 = 2w^2 - 4w
\]
Rearranging the equation gives:
\[
2w^2 - 4w - 96 = 0
\]
To simplify, we can divide the entire equation by 2:
\[
w^2 - 2w - 48 = 0
\]
Next, we can factor this quadratic equation. We need to find two numbers that multiply to \(-48\) and add to \(-2\). The numbers \(-8\) and \(6\) work here:
So, we can factor the equation as:
\[
(w - 8)(w + 6) = 0
\]
Setting each factor to zero gives us:
1. \( w - 8 = 0 \) which gives \( w = 8 \)
2. \( w + 6 = 0 \) which gives \( w = -6 \) (not a valid solution since width cannot be negative)
Thus, the width is:
\[
w = 8 \, \text{m}
\]
Now we can find the length using the expression for length in terms of width:
\[
l = 2(8) - 4 = 16 - 4 = 12 \, \text{m}
\]
So, the dimensions of the rectangle are:
**Length**: \( 12 \, \text{m} \)
**Width**: \( 8 \, \text{m} \)
To summarize:
- An expression for the length of the rectangle in terms of width is:
\[
l = 2w - 4
\]
- The formula for the area of a rectangle is:
\[
A = l \times w
\]
- Using trial and error, the values we found for the area of \( 96 \, \text{m}^2 \) yields dimensions:
\[
\text{Length} = 12 \, \text{m}, \quad \text{Width} = 8 \, \text{m}
\]