Yes, let's verify the solution step by step to ensure accuracy.
-
Expression for Length in Terms of Width:
Given that the length of the rectangle is four meters less than twice its width, we can express the length \( l \) in terms of the width \( w \):
\[
l = 2w - 4
\]
-
Formula for Area:
The area \( A \) of a rectangle is given by the formula:
\[
A = l \times w
\]
-
Setting Up the Equation:
We know that the area of the rectangle is 96 m². Substituting the expression for \( l \) into the area formula gives:
\[
A = (2w - 4) \times w
\]
\[
96 = (2w - 4) \times w
\]
Expanding the equation:
\[
96 = 2w^2 - 4w
\]
Rearranging gives:
\[
2w^2 - 4w - 96 = 0
\]
Dividing everything by 2 to simplify:
\[
w^2 - 2w - 48 = 0
\]
-
Factoring or Using Quadratic Formula:
We can factor this quadratic equation:
\[
(w - 8)(w + 6) = 0
\]
Thus, the solutions for \( w \) are:
\[
w = 8 \quad \text{or} \quad w = -6
\]
Since width cannot be negative, we have:
\[
w = 8 , \text{m}
\]
-
Finding the Length:
Now we substitute \( w = 8 \) back into the expression for \( l \):
\[
l = 2(8) - 4 = 16 - 4 = 12 , \text{m}
\]
-
Conclusion:
Therefore, the length and width of the rectangle are:
\[
l = 12 , \text{m}, \quad w = 8 , \text{m}
\]
So the answers originally provided are indeed correct. The final answers are:
- Expression for the length: \( l = 2w - 4 \)
- Formula for area: \( \text{area} = l \times w \)
- Length and width: \( l = 12 , \text{m} \) and \( w = 8 , \text{m} \)
1 answer