Breadth of a rectangular park is 12m and ratio of area of rectangular park to perimeter of rectangular park is, 42:11. If radius of circular park is equal to length of rectangular park and cost of fencing circular park is Rs.20/m, then find total cost of fencing circular park

Options

A. Rs.2250
B. Rs.2760
C. Rs.2800
D. Rs.2640

Answer

To solve the problem step by step, let's break it down:

### Step 1: Understand the Given Information
- The breadth of the rectangular park (b) = 12 m
- The ratio of the area of the rectangular park to its perimeter is 42:11.
- The radius of the circular park is equal to the length of the rectangular park (l).
- The cost of fencing the circular park is Rs. 20 per meter.

### Step 2: Set Up the Equations
1. **Area of the rectangular park (A)**:
\[ A = l \times b = l \times 12 \]

2. **Perimeter of the rectangular park (P)**:
\[ P = 2(l + b) = 2(l + 12) \]

3. **Ratio of Area to Perimeter**:
\[ \frac{A}{P} = \frac{42}{11} \]
Substituting the expressions for A and P:
\[ \frac{l \times 12}{2(l + 12)} = \frac{42}{11} \]

### Step 3: Cross Multiply to Solve for l
Cross multiplying gives:
\[ 11 \times (l \times 12) = 42 \times 2(l + 12) \]
This simplifies to:
\[ 11 \times 12l = 84(l + 12) \]
\[ 132l = 84l + 1008 \]
Subtracting \(84l\) from both sides:
\[ 132l - 84l = 1008 \]
\[ 48l = 1008 \]
Dividing both sides by 48:
\[ l = \frac{1008}{48} = 21 \text{ m} \]

### Step 4: Find the Radius of the Circular Park
Since the radius of the circular park is equal to the length of the rectangular park:
\[ r = l = 21 \text{ m} \]

### Step 5: Calculate the Circumference of the Circular Park
The circumference (C) of the circular park is given by:
\[ C = 2\pi r = 2 \times \frac{22}{7} \times 21 \]
Calculating this:
\[ C = \frac{44}{7} \times 21 = \frac{924}{7} = 132 \text{ m} \]

### Step 6: Calculate the Total Cost of Fencing the Circular Park
The total cost (Cost) for fencing is given by:
\[ \text{Cost} = \text{Circumference} \times \text{Cost per meter} \]
Substituting the values:
\[ \text{Cost} = 132 \times 20 = 2640 \text{ Rs} \]

### Final Answer
The total cost of fencing the circular park is Rs. 2640.

---