To find the area of the path alone around a circular garden, we need to follow a series of steps that involve calculating the areas of two circles and then subtracting the area of the smaller circle (the garden) from the area of the larger circle (the garden plus the path).
### Step-by-Step Solution
1. Determine the Radius of Both Circles:
- Garden Radius: The radius of the garden is given as [tex]\(8\)[/tex] feet.
- Total Radius (Garden + Path): The circular path adds an additional width of [tex]\(3\)[/tex] feet to the radius. Thus, the total radius is:
[tex]\[
\text{Total Radius} = \text{Garden Radius} + \text{Path Width} = 8 \, \text{ft} + 3 \, \text{ft} = 11 \, \text{ft}
\][/tex]
2. Calculate the Area of the Larger Circle (Garden + Path):
- We use the formula for the area of a circle, [tex]\(A = \pi r^2\)[/tex].
- Substitute [tex]\( \pi = 3.14 \)[/tex] and the total radius [tex]\(r = 11\)[/tex] feet:
[tex]\[
\text{Total Area} = 3.14 \times (11)^2 = 3.14 \times 121 = 379.94 \, \text{ft}^2
\][/tex]
3. Calculate the Area of the Smaller Circle (Garden):
- Again using the area formula [tex]\(A = \pi r^2\)[/tex], substitute [tex]\(\pi = 3.14\)[/tex] and the garden radius [tex]\(r = 8\)[/tex] feet:
[tex]\[
\text{Garden Area} = 3.14 \times (8)^2 = 3.14 \times 64 = 200.96 \, \text{ft}^2
\][/tex]
4. Determine the Area of the Path Alone:
- The area of the path is the difference between the area of the larger circle and the area of the smaller circle:
[tex]\[
\text{Path Area} = \text{Total Area} - \text{Garden Area} = 379.94 \, \text{ft}^2 - 200.96 \, \text{ft}^2 = 178.98 \, \text{ft}^2
\][/tex]
### Conclusion
The approximate area of the path alone is:
[tex]\[ 178.98 \, \text{ft}^2 \][/tex]
Thus, the correct answer is:
[tex]\[ \boxed{178.98 \, \text{ft}^2} \][/tex]
