A wire of length 44 cm is first bent to form a circle and then rebent to for a square. The difference of the two enclosed areas is

Options

A. `44 cm^(2)`
B. `33 cm^(2)`
C. `55 cm^(2)`
D. `66 cm^(2)`

Answer

To solve the problem step by step, we will first calculate the area of the circle formed by the wire and then the area of the square formed by the same wire. Finally, we will find the difference between the two areas.

### Step 1: Calculate the radius of the circle
The length of the wire is equal to the circumference of the circle when it is bent into a circle.

The formula for the circumference \( C \) of a circle is given by:
\[ C = 2\pi r \]
Where \( r \) is the radius of the circle.

Given that the length of the wire is 44 cm:
\[ 2\pi r = 44 \]

To find \( r \), we rearrange the equation:
\[ r = \frac{44}{2\pi} = \frac{22}{\pi} \]

### Step 2: 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 \( r \):
\[ A = \pi \left(\frac{22}{\pi}\right)^2 = \pi \cdot \frac{484}{\pi^2} = \frac{484}{\pi} \]

### Step 3: Calculate the side length of the square
When the wire is rebent to form a square, the length of the wire is equal to the perimeter \( P \) of the square.

The formula for the perimeter of a square is:
\[ P = 4s \]
Where \( s \) is the length of one side of the square.

Setting the perimeter equal to the length of the wire:
\[ 4s = 44 \]

To find \( s \):
\[ s = \frac{44}{4} = 11 \text{ cm} \]

### Step 4: Calculate the area of the square
The area \( A_s \) of a square is given by:
\[ A_s = s^2 \]

Substituting the value of \( s \):
\[ A_s = 11^2 = 121 \text{ cm}^2 \]

### Step 5: Calculate the difference of the two areas
Now we will find the difference between the area of the square and the area of the circle:
\[ \text{Difference} = A_s - A = 121 - \frac{484}{\pi} \]

### Final Result
The difference of the two enclosed areas is:
\[ \text{Difference} = 121 - \frac{484}{\pi} \text{ cm}^2 \]