Answer :
To determine the area of the path alone, we need to perform a series of calculations involving the circular garden and the surrounding path. Let's follow these steps:
1. Calculate the radius of the outer circle (garden + path):
- The radius of the garden is given as 8 feet.
- The width of the circular path is given as 3 feet.
- Therefore, the radius of the outer circle, which includes the garden and the path, is:
[tex]\[ \text{Outer radius} = 8 \text{ feet} + 3 \text{ feet} = 11 \text{ feet} \][/tex]
2. Calculate the area of the outer circle:
- The formula for the area of a circle is [tex]\(A = \pi r^2\)[/tex], where [tex]\(r\)[/tex] is the radius.
- Using [tex]\(\pi = 3.14 \)[/tex] and the outer radius of 11 feet:
[tex]\[ \text{Area of the outer circle} = 3.14 \times (11^2) = 3.14 \times 121 \approx 379.94 \text{ square feet} \][/tex]
3. Calculate the area of the inner circle (garden alone):
- The radius of the inner circle (garden) is 8 feet.
- Using the same area formula:
[tex]\[ \text{Area of the inner circle} = 3.14 \times (8^2) = 3.14 \times 64 \approx 200.96 \text{ square feet} \][/tex]
4. Calculate the area of the path alone:
- The area of the path is found by subtracting the area of the inner circle from the area of the outer circle:
[tex]\[ \text{Area of the path} = \text{Area of the outer circle} - \text{Area of the inner circle} \approx 379.94 \text{ square feet} - 200.96 \text{ square feet} \approx 178.98 \text{ square feet} \][/tex]
Given our calculations, the approximate area of the path alone is:
[tex]\[ \boxed{178.98 \text{ square feet}} \][/tex]
Hence, the correct answer is [tex]\(178.98 \, \text{ft}^2\)[/tex].
1. Calculate the radius of the outer circle (garden + path):
- The radius of the garden is given as 8 feet.
- The width of the circular path is given as 3 feet.
- Therefore, the radius of the outer circle, which includes the garden and the path, is:
[tex]\[ \text{Outer radius} = 8 \text{ feet} + 3 \text{ feet} = 11 \text{ feet} \][/tex]
2. Calculate the area of the outer circle:
- The formula for the area of a circle is [tex]\(A = \pi r^2\)[/tex], where [tex]\(r\)[/tex] is the radius.
- Using [tex]\(\pi = 3.14 \)[/tex] and the outer radius of 11 feet:
[tex]\[ \text{Area of the outer circle} = 3.14 \times (11^2) = 3.14 \times 121 \approx 379.94 \text{ square feet} \][/tex]
3. Calculate the area of the inner circle (garden alone):
- The radius of the inner circle (garden) is 8 feet.
- Using the same area formula:
[tex]\[ \text{Area of the inner circle} = 3.14 \times (8^2) = 3.14 \times 64 \approx 200.96 \text{ square feet} \][/tex]
4. Calculate the area of the path alone:
- The area of the path is found by subtracting the area of the inner circle from the area of the outer circle:
[tex]\[ \text{Area of the path} = \text{Area of the outer circle} - \text{Area of the inner circle} \approx 379.94 \text{ square feet} - 200.96 \text{ square feet} \approx 178.98 \text{ square feet} \][/tex]
Given our calculations, the approximate area of the path alone is:
[tex]\[ \boxed{178.98 \text{ square feet}} \][/tex]
Hence, the correct answer is [tex]\(178.98 \, \text{ft}^2\)[/tex].
