There is a path of uniform width 7 m round and outside a circular garden of diameter 210 m. Find the area of the path.

Answer

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

### Step 1: Find the radius of the circular garden
The diameter of the circular garden is given as 210 m. To find the radius, we use the formula:
\[ \text{Radius} = \frac{\text{Diameter}}{2} \]
So,
\[ R_1 = \frac{210}{2} = 105 \text{ m} \]

**Hint:** Remember that the radius is half of the diameter.

### Step 2: Find the radius of the outer circle
The path around the garden has a uniform width of 7 m. Therefore, the radius of the outer circle (which includes the path) is:
\[ R_2 = R_1 + \text{Width of the path} = 105 + 7 = 112 \text{ m} \]

**Hint:** To find the outer radius, simply add the width of the path to the inner radius.

### Step 3: Calculate the area of the outer circle
The area \(A\) of a circle is given by the formula:
\[ A = \pi r^2 \]
For the outer circle:
\[ A_{\text{outer}} = \pi R_2^2 = \pi (112)^2 \]

**Hint:** Use \( \pi \approx \frac{22}{7} \) for calculations if necessary.

### Step 4: Calculate the area of the inner circle
Using the same formula for the inner circle:
\[ A_{\text{inner}} = \pi R_1^2 = \pi (105)^2 \]

**Hint:** Remember to use the same value of \( \pi \) for both areas to ensure consistency.

### Step 5: Find the area of the path
The area of the path is the difference between the area of the outer circle and the area of the inner circle:
\[ A_{\text{path}} = A_{\text{outer}} - A_{\text{inner}} = \pi R_2^2 - \pi R_1^2 \]
This can be simplified to:
\[ A_{\text{path}} = \pi (R_2^2 - R_1^2) \]

### Step 6: Substitute the values and calculate
Substituting the values we found:
\[ A_{\text{path}} = \pi (112^2 - 105^2) \]
Calculating \(112^2\) and \(105^2\):
\[ 112^2 = 12544 \quad \text{and} \quad 105^2 = 11025 \]
Now, substituting these into the equation:
\[ A_{\text{path}} = \pi (12544 - 11025) = \pi (1520) \]
Now substituting \( \pi \approx \frac{22}{7} \):
\[ A_{\text{path}} = \frac{22}{7} \times 1520 \]

### Step 7: Final calculation
Calculating the area:
\[ A_{\text{path}} = \frac{22 \times 1520}{7} = \frac{33440}{7} \approx 4777.14 \text{ m}^2 \]

### Conclusion
The area of the path is approximately \(4777.14 \text{ m}^2\).

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