A path of width 3.5 m runs all around a circular pool having an area of `962.5 m^(2)`. Find the area of the path .

Answer

To find the area of the path that runs around a circular pool, we can follow these steps:

### Step 1: Find the radius of the circular pool
We know the area of the circular pool is given as \( 962.5 \, m^2 \). The formula for the area of a circle is:
\[ \text{Area} = \pi r^2 \]
Where \( r \) is the radius. We can rearrange this to find \( r^2 \):
\[ r^2 = \frac{\text{Area}}{\pi} \]
Substituting the values:
\[ r^2 = \frac{962.5}{\pi} \]
Using \( \pi \approx \frac{22}{7} \):
\[ r^2 = \frac{962.5 \times 7}{22} = \frac{6737.5}{22} = 306.25 \]

### Step 2: Calculate the radius \( r \)
Now, we take the square root of \( r^2 \) to find \( r \):
\[ r = \sqrt{306.25} = 17.5 \, m \]

### Step 3: Find the outer radius
The path has a width of \( 3.5 \, m \). Therefore, the outer radius \( R \) is:
\[ R = r + \text{width} = 17.5 + 3.5 = 21 \, m \]

### Step 4: Calculate the area of the outer circle
Using the outer radius \( R \):
\[ \text{Area of outer circle} = \pi R^2 = \pi (21^2) = \pi (441) \]

### Step 5: Calculate the area of the inner circle
Using the inner radius \( r \):
\[ \text{Area of inner circle} = \pi r^2 = \pi (17.5^2) = \pi (306.25) \]

### Step 6: Find the area of the path
The area of the path is the area of the outer circle minus the area of the inner circle:
\[ \text{Area of path} = \text{Area of outer circle} - \text{Area of inner circle} \]
\[ = \pi (441) - \pi (306.25) = \pi (441 - 306.25) = \pi (134.75) \]

### Step 7: Substitute \( \pi \) and calculate
Using \( \pi \approx \frac{22}{7} \):
\[ \text{Area of path} = \frac{22}{7} \times 134.75 \]
Calculating:
\[ = \frac{22 \times 134.75}{7} = \frac{2964.5}{7} \approx 423.5 \, m^2 \]

### Final Answer
The area of the path is approximately \( 423.5 \, m^2 \).

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