A wire is in the form of a circle of radius 42 m is cut and again bent in the form of a square . What is the diagonal of the square ?

Options

A. 66 m
B. `66 sqrt(3) m `
C. `66 sqrt(2) m`
D. none of these

Answer

To solve the problem step by step, we will follow these steps:

### Step 1: Calculate the circumference of the circle
The circumference \( C \) of a circle can be calculated using the formula:
\[ C = 2\pi r \]
where \( r \) is the radius of the circle. Given that the radius \( r = 42 \) m, we can substitute this value into the formula.

\[ C = 2 \times \pi \times 42 \]

### Step 2: Substitute the value of \( \pi \)
Using \( \pi \approx 3.14 \), we can calculate the circumference.

\[ C = 2 \times 3.14 \times 42 = 6.28 \times 42 \]

### Step 3: Perform the multiplication
Now, we will calculate \( 6.28 \times 42 \).

\[ C = 263.76 \text{ m} \]

### Step 4: Set the circumference equal to the perimeter of the square
When the wire is bent into the shape of a square, the length of the wire remains the same. Therefore, the perimeter \( P \) of the square is equal to the circumference of the circle.

\[ P = 4 \times \text{side of the square} \]

So we have:
\[ 4 \times \text{side} = 263.76 \]

### Step 5: Solve for the side of the square
To find the side of the square, we divide the perimeter by 4.

\[ \text{side} = \frac{263.76}{4} = 65.94 \text{ m} \]

### Step 6: Calculate the diagonal of the square
The diagonal \( d \) of a square can be calculated using the formula:
\[ d = \text{side} \times \sqrt{2} \]

Substituting the value of the side we found:

\[ d = 65.94 \times \sqrt{2} \]

### Step 7: Calculate the value of \( \sqrt{2} \)
Using \( \sqrt{2} \approx 1.414 \):

\[ d = 65.94 \times 1.414 \approx 93.33 \text{ m} \]

### Final Answer
Thus, the diagonal of the square is approximately \( 93.33 \) meters.

---