There is a circular garden of radius 21 metres. A path of width 3.5 metres is constructed just outside the garden. What is the area `("in meters"^(2))` of the path ?

Options

A. `500.5`
B. `575.6`
C. `521.2`
D. `560.7`

Answer

To find the area of the path surrounding the circular garden, we will follow these steps:

### Step-by-Step Solution:

1. **Identify the radius of the garden and the width of the path:**
- The radius of the circular garden (R1) = 21 meters.
- The width of the path = 3.5 meters.

2. **Calculate the radius of the larger circle (garden + path):**
- The radius of the larger circle (R2) = Radius of the garden + Width of the path
- R2 = R1 + 3.5 = 21 + 3.5 = 24.5 meters.

3. **Calculate the area of the larger circle (with the path):**
- Area of a circle = π * radius²
- Area of the larger circle = π * (R2)² = π * (24.5)².

4. **Calculate the area of the smaller circle (the garden):**
- Area of the smaller circle = π * (R1)² = π * (21)².

5. **Calculate the area of the path:**
- Area of the path = Area of the larger circle - Area of the smaller circle
- Area of the path = π * (24.5)² - π * (21)².

6. **Factor out π:**
- Area of the path = π * [(24.5)² - (21)²].

7. **Use the difference of squares formula:**
- a² - b² = (a - b)(a + b)
- Here, a = 24.5 and b = 21.
- Area of the path = π * [(24.5 - 21)(24.5 + 21)].

8. **Calculate the values:**
- (24.5 - 21) = 3.5
- (24.5 + 21) = 45.5
- Area of the path = π * (3.5 * 45.5).

9. **Calculate the numerical value:**
- Area of the path = π * (3.5 * 45.5) = π * 159.25.
- Using π ≈ 3.14, Area of the path ≈ 3.14 * 159.25 ≈ 500.5 m².

### Final Answer:
The area of the path is approximately **500.5 m²**.

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