A sector of a circle has an arc length of 2.4 feet and an area of 14.3 square feet. How many degrees are in the central angle?

Options

A. `63.4^(@)`
B. `20.2^(@)`
C. `14.3^(@)`
D. `11.5^(@)`

Explanation

To solve the problem step by step, we will use the formulas for arc length and area of a sector of a circle.

### Step 1: Understand the given information
We have:
- Arc length (s) = 2.4 feet
- Area of the sector (A) = 14.3 square feet

### Step 2: Write the formulas
1. The formula for arc length is:
\[ s = r \cdot \theta \]
where \( r \) is the radius and \( \theta \) is the central angle in radians.

2. The formula for the area of a sector is:
\[ A = \frac{1}{2} r^2 \cdot \theta \]

### Step 3: Set up the equations
From the arc length formula:
\[ 2.4 = r \cdot \theta \quad \text{(1)} \]
From the area formula:
\[ 14.3 = \frac{1}{2} r^2 \cdot \theta \quad \text{(2)} \]

### Step 4: Solve for \( r \) in terms of \( \theta \) from equation (1)
From equation (1):
\[ r = \frac{2.4}{\theta} \quad \text{(3)} \]

### Step 5: Substitute \( r \) from equation (3) into equation (2)
Substituting \( r \) into equation (2):
\[ 14.3 = \frac{1}{2} \left(\frac{2.4}{\theta}\right)^2 \cdot \theta \]
This simplifies to:
\[ 14.3 = \frac{1}{2} \cdot \frac{5.76}{\theta^2} \cdot \theta \]
\[ 14.3 = \frac{2.88}{\theta} \]

### Step 6: Solve for \( \theta \)
Multiplying both sides by \( \theta \):
\[ 14.3\theta = 2.88 \]
Now, divide both sides by 14.3:
\[ \theta = \frac{2.88}{14.3} \approx 0.2014 \text{ radians} \]

### Step 7: Convert \( \theta \) from radians to degrees
To convert radians to degrees, use the conversion factor:
\[ \text{Degrees} = \theta \cdot \frac{180}{\pi} \]
Substituting the value of \( \theta \):
\[ \text{Degrees} = 0.2014 \cdot \frac{180}{3.14} \approx 11.5 \text{ degrees} \]

### Final Answer
The central angle \( \theta \) is approximately **11.5 degrees**.

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