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We are given a rectangular garden with dimensions of 8 meters by 6 meters, and a pathway surrounding it. The task is to find the width of the pathway such that the larger rectangle formed (including the pathway) follows the Golden Ratio.

The Golden Ratio \( \phi \) is approximately 1.618034, and we are asked to use this value. Here's how we can approach the problem:

### 1. Setting up the dimensions:
- The original garden dimensions are 8 meters by 6 meters.
- Let the width of the pathway be \( x \) meters. So, the dimensions of the larger rectangle, which includes the pathway, will be:
  - Length: \( 8 + 2x \)
  - Width: \( 6 + 2x \)

### 2. Applying the Golden Ratio:
- The Golden Ratio between the length and width of the larger rectangle is given by:
  \[
  \frac{8 + 2x}{6 + 2x} = 1.618034
  \]
  
### 3. Solving for \( x \):
We'll solve this equation step by step.

\[
\frac{8 + 2x}{6 + 2x} = 1.618034
\]
Multiplying both sides by \( (6 + 2x) \) to get rid of the denominator:

\[
8 + 2x = 1.618034(6 + 2x)
\]

Now, we'll expand the right side:

\[
8 + 2x = 1.618034 \times 6 + 1.618034 \times 2x
\]
\[
8 + 2x = 9.708204 + 3.236068x
\]

Next, we subtract \( 2x \) from both sides:

\[
8 = 9.708204 + 1.236068x
\]

Now, subtract \( 9.708204 \) from both sides:

\[
8 - 9.708204 = 1.236068x
\]
\[
-1.708204 = 1.236068x
\]

Finally, divide both sides by 1.236068 to solve for \( x \):

\[
x = \frac{-1.708204}{1.236068}
\]

Let me compute the value of \( x \).The computed width of the pathway \( x \) comes out to be approximately \( -1.38 \) meters, which is not physically meaningful (the width of a pathway cannot be negative). This suggests there might be an issue in interpreting the dimensions in relation to the Golden Ratio.

To correct the issue, the garden's length may already follow the Golden Ratio relative to the pathway, or a different relationship may need to be considered.

Would you like further clarification on this, or for me to attempt an alternative approach to resolve the problem?

Here are 5 questions to explore this concept further:

1. Could there be a different interpretation of the problem involving the Golden Ratio?
2. Should we consider only the longer side of the garden when applying the ratio?
3. How can we account for negative values when solving geometric problems?
4. What are other real-world uses of the Golden Ratio in design?
5. How would changing the dimensions of the garden affect the width of the pathway?

**Tip:** When solving ratio-related problems, always double-check physical constraints like negative values to ensure real-world applicability.

A rectangular garden is to be made by a gardener with a dimension of 8m by 6m. It will be surrounded by a pathway. What is the width of the pathway such that the Golden Ratio is observed on the bigger rectangle made by adding the width of the pathway? Use the 1.618034 as the value of the Golden Ratio.

Learn how to determine the width of a pathway around a garden with dimensions 8m by 6m, such that the surrounding rectangle adheres to the Golden Ratio. The step-by-step solution involves setting up proportions and solving for the unknown variable x.

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