To solve this problem, you need to follow a series of steps. Let's break it down:
1. Identify the variables:
- Let the width of the rectangle be denoted as [tex]\( w \)[/tex] meters.
2. Express the length in terms of the width:
- According to the problem, the length [tex]\( l \)[/tex] is twice the width. Therefore, [tex]\( l = 2w \)[/tex].
3. Formulate the equation for the area:
- The area [tex]\( A \)[/tex] of a rectangle is calculated by multiplying its length and its width. According to the problem, the area is 128 square meters.
- Thus, [tex]\( A = l \times w \)[/tex].
- Substituting the given values, we get:
[tex]\[
128 = 2w \times w
\][/tex]
4. Solve for the width [tex]\( w \)[/tex]:
- Simplify the equation:
[tex]\[
128 = 2w^2
\][/tex]
- Divide both sides of the equation by 2:
[tex]\[
64 = w^2
\][/tex]
- Take the square root of both sides to find [tex]\( w \)[/tex]:
[tex]\[
w = \sqrt{64}
\][/tex]
[tex]\[
w = 8
\][/tex]
5. Determine the length [tex]\( l \)[/tex]:
- Remember that the length [tex]\( l \)[/tex] is twice the width:
[tex]\[
l = 2w = 2 \times 8 = 16
\][/tex]
6. Calculate the perimeter:
- The perimeter [tex]\( P \)[/tex] of a rectangle is given by the formula:
[tex]\[
P = 2 \times (l + w)
\][/tex]
- Substitute the values of [tex]\( l \)[/tex] and [tex]\( w \)[/tex]:
[tex]\[
P = 2 \times (16 + 8)
\][/tex]
[tex]\[
P = 2 \times 24
\][/tex]
[tex]\[
P = 48
\][/tex]
Therefore, the width of the rectangle is 8 meters, the length is 16 meters, and the perimeter is 48 meters.
