Assertion (A) : If a wire of length 22 cm is bent in the shape of a circle, then area of the circle so formed is `40 cm^(2)`
Reason (R) : Circumference of the circle = length of the wire

Options

A. Both A and R are true and R is the correct explanation of A
B. Both A and R are true and R is not correct explanation of A
C. A is true but R is false
D. A is false but R is true

Answer

To solve the problem step by step, we will analyze the assertion and the reason provided in the question.

### Step 1: Understand the Assertion and Reason
- **Assertion (A)**: If a wire of length 22 cm is bent in the shape of a circle, then the area of the circle so formed is 40 cm².
- **Reason (R)**: The circumference of the circle = length of the wire.

### Step 2: Establish the Relationship between Length of Wire and Circumference
The length of the wire is given as 22 cm. When the wire is bent into the shape of a circle, this length becomes the circumference of the circle.

**Formula for Circumference of a Circle**:
\[ C = 2\pi r \]
Where \( C \) is the circumference and \( r \) is the radius.

### Step 3: Set Up the Equation
Since the circumference of the circle is equal to the length of the wire, we have:
\[ 2\pi r = 22 \]

### Step 4: Solve for the Radius (r)
To find the radius \( r \), we can rearrange the equation:
\[ r = \frac{22}{2\pi} = \frac{11}{\pi} \]

Using the approximation \( \pi \approx \frac{22}{7} \):
\[ r = \frac{11}{\frac{22}{7}} = \frac{11 \times 7}{22} = \frac{77}{22} = \frac{7}{2} \text{ cm} \]

### Step 5: Calculate the Area of the Circle
The area \( A \) of a circle is given by the formula:
\[ A = \pi r^2 \]

Substituting the value of \( r \):
\[ A = \pi \left(\frac{7}{2}\right)^2 = \pi \cdot \frac{49}{4} = \frac{49\pi}{4} \]

Using \( \pi \approx \frac{22}{7} \):
\[ A = \frac{49 \times \frac{22}{7}}{4} = \frac{49 \times 22}{28} = \frac{1078}{28} = 38.5 \text{ cm}^2 \]

### Step 6: Compare the Area with the Assertion
The calculated area of the circle is 38.5 cm², which is not equal to the assertion that it is 40 cm². Therefore, the assertion is false.

### Step 7: Evaluate the Reason
The reason states that the circumference of the circle is equal to the length of the wire, which is true.

### Conclusion
- **Assertion (A)**: False (the area is 38.5 cm², not 40 cm²).
- **Reason (R)**: True (the circumference equals the length of the wire).

Thus, the final answer is that the assertion is false, but the reason is true.

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