Ravi cuts a square field of maximum possible area from his circular field. Find the area of the remaining circular field if the cost of grass cutting of the square field at Rs 4 per `m^(2)` is Rs 882?

Options

A. `120 m ^(2)`
B. `126 m ^(2)`
C. `121 m ^(2)`
D. `116 m ^(2)`

Answer

To solve the problem step by step, we will break down the information given and calculate the required areas.

### Step 1: Determine the area of the square field
We know that the cost of cutting grass in the square field is Rs 882, and the cost per square meter is Rs 4.

To find the area of the square field, we can use the formula:

\[ \text{Area of square} = \frac{\text{Total cost}}{\text{Cost per m}^2} \]

Substituting the values:

\[ \text{Area of square} = \frac{882}{4} = 220.5 \, m^2 \]

### Step 2: Find the side length of the square
The area of a square is given by the formula:

\[ \text{Area} = \text{side}^2 \]

Let the side length of the square be \( s \). Therefore, we have:

\[ s^2 = 220.5 \]

To find \( s \), we take the square root:

\[ s = \sqrt{220.5} \approx 14.85 \, m \]

### Step 3: Determine the radius of the circular field
The maximum square that can be cut from a circle will have its diagonal equal to the diameter of the circle. The diagonal \( d \) of the square can be calculated using the formula:

\[ d = s\sqrt{2} \]

Substituting the value of \( s \):

\[ d = 14.85 \sqrt{2} \approx 21.0 \, m \]

Since the diameter of the circle is equal to the diagonal of the square, we can find the radius \( r \) of the circular field:

\[ r = \frac{d}{2} = \frac{21.0}{2} \approx 10.5 \, m \]

### 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 \( r \):

\[ A = \pi (10.5)^2 \approx \pi \times 110.25 \approx 346.36 \, m^2 \]

### Step 5: Calculate the area of the remaining circular field
To find the area of the remaining circular field, we subtract the area of the square field from the area of the circular field:

\[ \text{Remaining area} = \text{Area of circle} - \text{Area of square} \]

Substituting the values:

\[ \text{Remaining area} = 346.36 - 220.5 \approx 125.86 \, m^2 \]

### Final Answer
The area of the remaining circular field is approximately **125.86 m²**.

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