Let the radius of the circular garden be $r$, and the width of the path be $w$.

Step 1: Garden Area Equation

The area of the circular garden is given by:

$A_{\text{garden}} = \pi r^2 = 254.34 \, \text{m}^2$

Solving for $r$:

$r^2 = \frac{254.34}{\pi}$

$r = \sqrt{\frac{254.34}{\pi}}$

Step 2: Total Area Equation

The total area of the garden plus the path is:

$A_{\text{total}} = \pi (r + w)^2 = 314.0 \, \text{m}^2$

Now, we have two equations:

1. $\pi r^2 = 254.34$
2. $\pi (r + w)^2 = 314.0$

Step 3: Solving for $w$

First, calculate $r$, then substitute it into the total area equation and solve for $w$. Let's calculate the exact values now.The radius of the circular garden is approximately $9.00 \, \text{m}$, and the width of the path is approximately $1.00 \, \text{m}$.

Would you like further details or have any questions?

Here are some related questions for further exploration:

1. How does the width of the path affect the total area of the circular garden?
2. If the width of the path is doubled, what would the total area become?
3. How would the result change if the garden had a different initial area?
4. What is the formula for the circumference of the garden including the path?
5. How could we apply this problem to find the cost of paving the path if the price per square meter is known?

Tip: When dealing with geometric shapes, always break the problem down by focusing first on the basic shape before including any additions like paths or borders.