If a circle has a circumference of 16 inches, the area of a sector with a central angle of 4.7 radians is

Options

A. 10
B. 12
C. 15
D. 25

Explanation

To find the area of a sector with a central angle of 4.7 radians in a circle with a circumference of 16 inches, we can follow these steps:

### Step 1: Find the radius of the circle
The formula for the circumference \( C \) of a circle is given by:
\[ C = 2\pi r \]
where \( r \) is the radius. We know the circumference is 16 inches:
\[ 16 = 2\pi r \]
To find \( r \), we can rearrange the equation:
\[ r = \frac{16}{2\pi} = \frac{8}{\pi} \]
Calculating this gives:
\[ r \approx \frac{8}{3.14} \approx 2.55 \text{ inches} \]

### Step 2: Use the formula for the area of a sector
The area \( A \) of a sector with a central angle \( \theta \) (in radians) is given by the formula:
\[ A = \frac{1}{2} r^2 \theta \]
Substituting the values we have:
\[ A = \frac{1}{2} \left(\frac{8}{\pi}\right)^2 \cdot 4.7 \]

### Step 3: Calculate \( r^2 \)
First, we calculate \( r^2 \):
\[ r^2 = \left(\frac{8}{\pi}\right)^2 = \frac{64}{\pi^2} \]

### Step 4: Substitute \( r^2 \) into the area formula
Now substituting \( r^2 \) into the area formula:
\[ A = \frac{1}{2} \cdot \frac{64}{\pi^2} \cdot 4.7 \]

### Step 5: Simplify the expression
Calculating this gives:
\[ A = \frac{64 \cdot 4.7}{2\pi^2} = \frac{300.8}{2\pi^2} = \frac{150.4}{\pi^2} \]

### Step 6: Calculate the numerical value
Using \( \pi \approx 3.14 \):
\[ A \approx \frac{150.4}{(3.14)^2} \approx \frac{150.4}{9.8596} \approx 15.24 \text{ square inches} \]

Thus, the area of the sector is approximately:
\[ A \approx 15 \text{ square inches} \]

### Final Answer
The area of the sector with a central angle of 4.7 radians is approximately **15 square inches**.

---