The area of the ring between two concentric circles, whose circumference are 88 cm and 132 cm, is: 

Options

A. `780 cm^(2)`
B. `770 cm^(2)`
C. `715 cm^(2)`
D. `660 cm^(2)`

Answer

To find the area of the ring between two concentric circles with given circumferences, we can follow these steps:

### Step 1: Find the radius of the larger circle
The circumference \( C \) of a circle is given by the formula:
\[ C = 2\pi R \]
where \( R \) is the radius of the circle. For the larger circle with a circumference of 132 cm:
\[ 132 = 2\pi R \]
To find \( R \), we rearrange the equation:
\[ R = \frac{132}{2\pi} = \frac{132}{2 \times \frac{22}{7}} = \frac{132 \times 7}{44} = \frac{924}{44} = 21 \text{ cm} \]

### Step 2: Find the radius of the smaller circle
Using the same formula for the smaller circle with a circumference of 88 cm:
\[ 88 = 2\pi r \]
Rearranging gives:
\[ r = \frac{88}{2\pi} = \frac{88}{2 \times \frac{22}{7}} = \frac{88 \times 7}{44} = \frac{616}{44} = 14 \text{ cm} \]

### Step 3: Calculate the area of the larger circle
The area \( A \) of a circle is given by:
\[ A = \pi R^2 \]
For the larger circle:
\[ A_{\text{large}} = \pi (21)^2 = \pi \times 441 = \frac{22}{7} \times 441 = 1386 \text{ cm}^2 \]

### Step 4: Calculate the area of the smaller circle
For the smaller circle:
\[ A_{\text{small}} = \pi (14)^2 = \pi \times 196 = \frac{22}{7} \times 196 = 616 \text{ cm}^2 \]

### Step 5: Calculate the area of the ring
The area of the ring is the difference between the area of the larger circle and the area of the smaller circle:
\[ A_{\text{ring}} = A_{\text{large}} - A_{\text{small}} = 1386 - 616 = 770 \text{ cm}^2 \]

### Final Answer
The area of the ring between the two concentric circles is:
\[ \boxed{770 \text{ cm}^2} \]

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