Let's go through the problem step-by-step to find the area of the path alone.
First, we need to determine the radius of the entire circular area, which includes both the garden and the path. The radius of the garden is 8 feet, and the width of the path is 3 feet. Therefore, the total radius (radius of the garden plus the width of the path) is:
[tex]\[ 8 \, \text{feet} + 3 \, \text{feet} = 11 \, \text{feet} \][/tex]
Next, we calculate the area of the garden itself. The formula for the area of a circle is:
[tex]\[ \text{Area} = \pi \times \text{radius}^2 \][/tex]
For the garden:
[tex]\[ \text{Area of the garden} = 3.14 \times (8 \, \text{feet})^2 = 3.14 \times 64 = 200.96 \, \text{square feet} \][/tex]
Then, we calculate the area of the entire circular space that includes both the garden and the path using the total radius of 11 feet:
[tex]\[ \text{Area of the entire circular space} = 3.14 \times (11 \, \text{feet})^2 = 3.14 \times 121 = 379.94 \, \text{square feet} \][/tex]
To find the area of the path alone, we subtract the area of the garden from the area of the entire circular space:
[tex]\[ \text{Area of the path} = 379.94 \, \text{square feet} - 200.96 \, \text{square feet} = 178.98 \, \text{square feet} \][/tex]
Therefore, the approximate area of the path alone is:
[tex]\[
\boxed{178.98 \, \text{square feet}}
\][/tex]
