The circumference of a circle is 11 cm and the angle of a sector of the circle is `60^(@)`. The area of the sector is (use `pi = (22)/(7)`)

Options

A. `1(29)/(48) cm^(2)`
B. `2(29)/(48) cm^(2)`
C. `1(27)/(48) cm^(2)`
D. `2(27)/(48) cm^(2)`

Answer

To solve the problem, we need to find the area of the sector of a circle given the circumference and the angle of the sector. Here are the steps to find the solution:

### Step 1: Find the radius of the circle
The formula for the circumference (C) of a circle is given by:
\[ C = 2\pi r \]
We know that the circumference is 11 cm. Therefore, we can set up the equation:
\[ 2\pi r = 11 \]

Substituting \(\pi\) with \(\frac{22}{7}\):
\[ 2 \times \frac{22}{7} \times r = 11 \]

Now, simplify this equation to find \(r\):
\[ \frac{44}{7} r = 11 \]

To isolate \(r\), multiply both sides by \(\frac{7}{44}\):
\[ r = 11 \times \frac{7}{44} \]
\[ r = \frac{77}{44} = \frac{7}{4} \text{ cm} \]

### Step 2: Use the formula for the area of the sector
The formula for the area (A) of a sector of a circle is given by:
\[ A = \frac{\theta}{360} \times \pi r^2 \]
where \(\theta\) is the angle of the sector in degrees.

Given \(\theta = 60^\circ\) and \(r = \frac{7}{4}\):
\[ A = \frac{60}{360} \times \frac{22}{7} \times \left(\frac{7}{4}\right)^2 \]

### Step 3: Simplify the area calculation
First, simplify \(\frac{60}{360}\):
\[ \frac{60}{360} = \frac{1}{6} \]

Now, calculate \(r^2\):
\[ \left(\frac{7}{4}\right)^2 = \frac{49}{16} \]

Now substitute these values back into the area formula:
\[ A = \frac{1}{6} \times \frac{22}{7} \times \frac{49}{16} \]

### Step 4: Calculate the area
Now, multiply the fractions:
\[ A = \frac{22 \times 49}{6 \times 7 \times 16} \]

Calculating the numerator:
\[ 22 \times 49 = 1078 \]

Calculating the denominator:
\[ 6 \times 7 \times 16 = 672 \]

Thus, we have:
\[ A = \frac{1078}{672} \]

### Step 5: Simplify the fraction
We can simplify \(\frac{1078}{672}\):
Both numbers can be divided by 2:
\[ A = \frac{539}{336} \]

### Final Answer
The area of the sector is:
\[ A \approx 1.604 \text{ cm}^2 \]