The circumference of two circles is 44 metres and 440 metres respectively. What is the difference between the area of the larger circle and the smaller circle ?

Options

A. 16245 sq mt
B. 15246 sq.mt
C. 12546 sq.mt
D. 12645 sq.mt

Answer

To find the difference between the area of the larger circle and the smaller circle given their circumferences, we can follow these steps:

### Step 1: Identify the circumferences of the circles
Let the circumference of the smaller circle \( C_1 = 44 \) metres and the circumference of the larger circle \( C_2 = 440 \) metres.

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

For the smaller circle:
\[ C_1 = 2\pi r_1 \]
Substituting the value of \( C_1 \):
\[ 44 = 2\pi r_1 \]
To find \( r_1 \):
\[ r_1 = \frac{44}{2\pi} = \frac{44}{2 \times \frac{22}{7}} = \frac{44 \times 7}{44} = 7 \text{ metres} \]

For the larger circle:
\[ C_2 = 2\pi r_2 \]
Substituting the value of \( C_2 \):
\[ 440 = 2\pi r_2 \]
To find \( r_2 \):
\[ r_2 = \frac{440}{2\pi} = \frac{440}{2 \times \frac{22}{7}} = \frac{440 \times 7}{44} = 70 \text{ metres} \]

### Step 3: Calculate the areas of both circles
The area \( A \) of a circle is given by:
\[ A = \pi r^2 \]

For the smaller circle:
\[ A_1 = \pi r_1^2 = \pi (7^2) = \pi \times 49 \]

For the larger circle:
\[ A_2 = \pi r_2^2 = \pi (70^2) = \pi \times 4900 \]

### Step 4: Find the difference between the areas
The difference in area \( A_2 - A_1 \) is:
\[ A_2 - A_1 = \pi \times 4900 - \pi \times 49 = \pi (4900 - 49) = \pi \times 4851 \]

### Step 5: Substitute the value of \( \pi \) to find the numerical difference
Using \( \pi \approx \frac{22}{7} \):
\[ A_2 - A_1 = \frac{22}{7} \times 4851 \]
Calculating this:
\[ = \frac{22 \times 4851}{7} = \frac{106772}{7} \approx 15253.14 \text{ square metres} \]

### Final Result
The difference between the area of the larger circle and the smaller circle is approximately \( 15253.14 \) square metres.

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