A circular park is surrounded by a road 28 m wide. Find the area of the road if the circumference of the park is 880 m.

Answer

To find the area of the road surrounding the circular park, we can follow these steps:

### Step 1: Find the radius of the park
We know the circumference of the park is given by the formula:
\[ C = 2\pi r \]
where \(C\) is the circumference and \(r\) is the radius. Given that the circumference is 880 m, we can set up the equation:
\[ 2\pi r = 880 \]
Using \(\pi \approx \frac{22}{7}\), we can substitute and solve for \(r\):
\[ 2 \times \frac{22}{7} \times r = 880 \]
\[ \frac{44}{7} r = 880 \]
Multiplying both sides by \(\frac{7}{44}\):
\[ r = 880 \times \frac{7}{44} \]
\[ r = 20 \times 7 = 140 \text{ m} \]

### Step 2: Find the outer radius
The width of the road is given as 28 m. Therefore, the outer radius \(R\) is:
\[ R = r + \text{width of the road} = 140 \text{ m} + 28 \text{ m} = 168 \text{ m} \]

### Step 3: Calculate the area of the outer circle
The area \(A_{outer}\) of the outer circle is given by:
\[ A_{outer} = \pi R^2 \]
Substituting \(R = 168\) m:
\[ A_{outer} = \pi (168)^2 \]
Calculating \(168^2\):
\[ 168^2 = 28224 \]
Thus,
\[ A_{outer} = \pi \times 28224 \]

### Step 4: Calculate the area of the inner circle
The area \(A_{inner}\) of the inner circle (the park) is given by:
\[ A_{inner} = \pi r^2 \]
Substituting \(r = 140\) m:
\[ A_{inner} = \pi (140)^2 \]
Calculating \(140^2\):
\[ 140^2 = 19600 \]
Thus,
\[ A_{inner} = \pi \times 19600 \]

### Step 5: Find the area of the road
The area of the road is the difference between the area of the outer circle and the area of the inner circle:
\[ A_{road} = A_{outer} - A_{inner} \]
Substituting the areas:
\[ A_{road} = \pi \times 28224 - \pi \times 19600 \]
Factoring out \(\pi\):
\[ A_{road} = \pi (28224 - 19600) \]
Calculating the difference:
\[ 28224 - 19600 = 8624 \]
Thus,
\[ A_{road} = \pi \times 8624 \]
Substituting \(\pi \approx \frac{22}{7}\):
\[ A_{road} = \frac{22}{7} \times 8624 \]
Calculating:
\[ A_{road} = \frac{22 \times 8624}{7} = \frac{189728}{7} = 27104 \text{ m}^2 \]

### Final Answer:
The area of the road is \(27104 \text{ m}^2\).

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