The cost of fencing a circular field at the rate of Rs 24 per metre is Rs 5280. The field is to be ploughed at the rate of `R s\ 0. 50\ p e r\ m^2`. Find the cost of ploughing the field .

Explanation

To solve the problem step by step, we will follow the logical sequence outlined in the video transcript.

### Step 1: Calculate the Circumference of the Circular Field
The cost of fencing the circular field is given as Rs 5280, and the rate of fencing is Rs 24 per meter.

To find the circumference (C) of the circular field, we use the formula:
\[ C = \frac{\text{Cost}}{\text{Rate}} \]
Substituting the values:
\[ C = \frac{5280}{24} \]
Calculating this gives:
\[ C = 220 \text{ meters} \]

### Step 2: Find the Radius of the Circular Field
The circumference of a circle is also given by the formula:
\[ C = 2\pi r \]
Where \( r \) is the radius. We can rearrange this formula to find \( r \):
\[ r = \frac{C}{2\pi} \]
Substituting the value of \( C \) and using \( \pi \approx \frac{22}{7} \):
\[ r = \frac{220}{2 \times \frac{22}{7}} \]
Calculating this step by step:
1. Calculate \( 2 \times \frac{22}{7} = \frac{44}{7} \)
2. Now, \( r = \frac{220 \times 7}{44} \)
3. Simplifying \( \frac{220}{44} = 5 \)
4. Therefore, \( r = 5 \times 7 = 35 \text{ meters} \)

### Step 3: 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 \( r \):
\[ A = \frac{22}{7} \times (35)^2 \]
Calculating \( (35)^2 = 1225 \):
\[ A = \frac{22}{7} \times 1225 \]
Now, simplifying:
1. \( \frac{1225}{7} = 175 \)
2. Therefore, \( A = 22 \times 175 = 3850 \text{ m}^2 \)

### Step 4: Calculate the Cost of Ploughing the Field
The cost of ploughing is given as Rs 0.50 per square meter. To find the total cost (C_p) of ploughing, we use the formula:
\[ C_p = \text{Area} \times \text{Rate} \]
Substituting the values:
\[ C_p = 3850 \times 0.50 \]
Calculating this gives:
\[ C_p = 1925 \text{ rupees} \]

### Final Answer
The cost of ploughing the field is Rs 1925.

---