- Identify the inner radius \(r_1 = 8 \text{ ft}\) and outer radius \(r_2 = 11 \text{ ft}\).
- Calculate the area of the garden (\(200.96 \text{ ft}^2\)) and the total area including the path (\(379.94 \text{ ft}^2\)) using \(\pi \approx 3.14\).
- Subtract the garden area from the total area to find the path's area.
- The area of the path is \(\boxed{178.98 \text{ ft}^2}\).
Explanation
- Define dimensions of the garden and the path.
The problem asks for the area of a circular path surrounding a circular garden.
The garden has a radius of \(r_1 = 8 \text{ ft}\).
The path has a constant width of \(w = 3 \text{ ft}\), meaning the path extends outward from the edge of the garden.
The outer boundary of the path forms a larger circle concentric with the garden.
The radius of this outer circle is the sum of the garden's radius and the path's width:
\(r_2 = r_1 + w = 8 \text{ ft} + 3 \text{ ft} = 11 \text{ ft}\).
- Calculate the areas of the inner and outer circles.
To find the area of the path, we need to subtract the area of the inner circle (the garden) from the area of the entire outer circle (garden plus path).
The area of a circle is given by the formula \(A = \pi r^2\).
Using \(\pi \approx 3.14\):
- The area of the garden is \(A_{\text{garden}} = \pi \times r_1^2 = 3.14 \times 8^2 = 3.14 \times 64 = 200.96 \text{ ft}^2\).
- The total area of the garden plus the path is \(A_{\text{total}} = \pi \times r_2^2 = 3.14 \times 11^2 = 3.14 \times 121 = 379.94 \text{ ft}^2\).
- Calculate the final area of the path.
The area of the path alone is the difference between the total area and the area of the garden:
\(A_{\text{path}} = A_{\text{total}} - A_{\text{garden}} = 379.94 \text{ ft}^2 - 200.96 \text{ ft}^2 = 178.98 \text{ ft}^2\).
Alternatively, this can be calculated as:
\(A_{\text{path}} = \pi (r_2^2 - r_1^2) = 3.14 \times (11^2 - 8^2) = 3.14 \times (121 - 64) = 3.14 \times 57 = 178.98 \text{ ft}^2\).
The approximate area of the path is \(\boxed{178.98 \text{ ft}^2}\).
Examples
Imagine you are a landscape designer planning a circular rose garden. You want to install a decorative stone path of uniform width around the roses to allow for easy maintenance and viewing. By calculating the area of this path, you can accurately estimate how many paving stones or bags of mulch are needed to cover the entire pathway, ensuring your project stays within budget and uses materials efficiently. This simple application of geometry helps bridge the gap between a design concept and its real-world implementation.