Let's solve this problem step-by-step:
1. Determine the initial dimensions of the rectangular plot:
- Length (L1) = 30 feet
- Width (W1) = 20 feet
2. Compute the initial area of the rectangle:
[tex]\[
\text{Initial Area} = L1 \times W1 = 30 \times 20 = 600 \text{ square feet}
\][/tex]
3. Determine the target area (which is double the initial area):
[tex]\[
\text{Target Area} = 2 \times \text{Initial Area} = 2 \times 600 = 1200 \text{ square feet}
\][/tex]
4. Introduce the variable `x` to represent the width of the border around the plot.
5. Find the new dimensions of the rectangle after adding the border:
- The border increases both the length and width by `2x` (since the border is on both sides of each dimension).
- New Length (L2) = [tex]\(30 + 2x\)[/tex]
- New Width (W2) = [tex]\(20 + 2x\)[/tex]
6. Express the new area of the rectangle including the border:
[tex]\[
\text{New Area} = (L2) \times (W2) = (30 + 2x) \times (20 + 2x)
\][/tex]
7. Set up the equation to reflect the condition that the new area must equal the target area:
[tex]\[
(30 + 2x) \times (20 + 2x) = 1200
\][/tex]
So, the equation that can be used to find the width of the border [tex]\(x\)[/tex] is:
[tex]\[
(30 + 2x) \times (20 + 2x) = 1200
\][/tex]
This equation embodies the relationship required to solve for the width of the border `x` that would double the area of the original rectangular plot.
