To fence a circular field, the total cost is `Rs. 26400` at the rate of `Rs. 50` per metre. Find the cost of ploughing the field at the rate of `Rs. 10` per metre square.

Options

A. `Rs. 221760`
B. `Rs . 154740`
C. `Rs. 212706`
D. `Rs. 202500`

Answer

To solve the problem, we need to follow these steps:

### Step 1: Calculate the total length of the fence
The total cost of fencing the circular field is given as Rs. 26400, and the cost per meter is Rs. 50. We can find the total length of the fence by dividing the total cost by the cost per meter.

\[ \text{Total length of the fence} = \frac{\text{Total cost}}{\text{Cost per meter}} = \frac{26400}{50} \]

### Step 2: Perform the calculation
Now, we perform the division:

\[ \text{Total length of the fence} = \frac{26400}{50} = 528 \text{ meters} \]

### Step 3: Calculate the radius of the circular field
The circumference of a circle is given by the formula:

\[ C = 2\pi r \]

Where \(C\) is the circumference and \(r\) is the radius. We can rearrange this formula to find the radius:

\[ r = \frac{C}{2\pi} \]

Substituting the value of \(C\):

\[ r = \frac{528}{2\pi} \]

### Step 4: Calculate the area of the circular field
The area \(A\) of a circle is given by the formula:

\[ A = \pi r^2 \]

Substituting the value of \(r\):

\[ A = \pi \left(\frac{528}{2\pi}\right)^2 \]

### Step 5: Simplify the area calculation
Calculating \(r^2\):

\[ r^2 = \left(\frac{528}{2\pi}\right)^2 = \frac{528^2}{4\pi^2} \]

Thus, the area becomes:

\[ A = \pi \cdot \frac{528^2}{4\pi^2} = \frac{528^2}{4\pi} \]

### Step 6: Calculate the cost of ploughing the field
The cost of ploughing is given at the rate of Rs. 10 per square meter. Therefore, the total cost of ploughing the field is:

\[ \text{Cost of ploughing} = \text{Area} \times \text{Cost per square meter} \]

Substituting the area:

\[ \text{Cost of ploughing} = \frac{528^2}{4\pi} \times 10 \]

### Step 7: Perform the final calculation
Calculating \(528^2\):

\[ 528^2 = 278784 \]

Now substituting back:

\[ \text{Cost of ploughing} = \frac{278784}{4\pi} \times 10 = \frac{2787840}{4\pi} \]

Using \(\pi \approx 3.14\):

\[ \text{Cost of ploughing} = \frac{2787840}{4 \times 3.14} \approx \frac{2787840}{12.56} \approx 221,000 \]

Thus, the cost of ploughing the field is approximately Rs. 221,000.