A circular swimming pool is surrounded by a concrete wall 4m wide. If the area of the concrete wall surrounding the pool is `(11)/(25)` that of the pool, then the radius (in m) of the pool is :

Options

A. 8
B. 6
C. 30
D. 20

Answer

To solve the problem step by step, we will follow the mathematical reasoning outlined in the video transcript.

### Step 1: Understand the Problem
We have a circular swimming pool with a radius \( r \) meters, surrounded by a concrete wall that is 4 meters wide. We need to find the radius of the pool given that the area of the concrete wall is \( \frac{11}{25} \) times the area of the pool.

### Step 2: Write Down the Areas
1. **Area of the pool**:
\[ A_{\text{pool}} = \pi r^2 \]

2. **Radius of the pool including the wall**:
\[ R = r + 4 \]

3. **Area of the pool including the wall**:
\[ A_{\text{total}} = \pi R^2 = \pi (r + 4)^2 \]

4. **Area of the concrete wall**:
\[ A_{\text{wall}} = A_{\text{total}} - A_{\text{pool}} = \pi (r + 4)^2 - \pi r^2 \]

### Step 3: Simplify the Area of the Wall
Using the difference of squares:
\[ A_{\text{wall}} = \pi \left[(r + 4)^2 - r^2\right] \]
\[ = \pi \left[r^2 + 8r + 16 - r^2\right] \]
\[ = \pi (8r + 16) \]

### Step 4: Set Up the Equation
According to the problem, the area of the wall is \( \frac{11}{25} \) times the area of the pool:
\[ \pi (8r + 16) = \frac{11}{25} \pi r^2 \]

### Step 5: Cancel \( \pi \) and Rearrange
Cancelling \( \pi \) from both sides:
\[ 8r + 16 = \frac{11}{25} r^2 \]
Multiplying through by 25 to eliminate the fraction:
\[ 25(8r + 16) = 11r^2 \]
\[ 200r + 400 = 11r^2 \]
Rearranging gives:
\[ 11r^2 - 200r - 400 = 0 \]

### Step 6: Solve the Quadratic Equation
Using the quadratic formula \( r = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \):
- Here, \( a = 11 \), \( b = -200 \), and \( c = -400 \).
- Calculate the discriminant:
\[ b^2 - 4ac = (-200)^2 - 4 \cdot 11 \cdot (-400) = 40000 + 17600 = 57600 \]
- Now, apply the quadratic formula:
\[ r = \frac{200 \pm \sqrt{57600}}{22} \]
\[ \sqrt{57600} = 240 \]
\[ r = \frac{200 \pm 240}{22} \]

Calculating the two possible values:
1. \( r = \frac{440}{22} = 20 \)
2. \( r = \frac{-40}{22} \) (not valid as radius cannot be negative)

### Final Answer
Thus, the radius of the pool is:
\[ \boxed{20 \text{ m}} \]