If a sector of a circle has an arc length of `2pi` inches and an area of `6pi` square inches, what is the length of the radius of the circle?

Options

A. 1
B. 2
C. 3
D. 6

Explanation

To find the length of the radius of the circle given the arc length and area of the sector, we can use the following steps:

### Step 1: Write down the formulas for arc length and area of a sector.
The arc length \( s \) of a sector is given by the formula:
\[ s = r \theta \]
where \( r \) is the radius and \( \theta \) is the angle in radians.

The area \( A \) of a sector is given by the formula:
\[ A = \frac{1}{2} r^2 \theta \]

### Step 2: Substitute the given values into the formulas.
From the problem, we know:
- Arc length \( s = 2\pi \) inches
- Area \( A = 6\pi \) square inches

Using the arc length formula:
\[ 2\pi = r \theta \quad \text{(1)} \]

Using the area formula:
\[ 6\pi = \frac{1}{2} r^2 \theta \quad \text{(2)} \]

### Step 3: Solve equation (1) for \( \theta \).
From equation (1):
\[ \theta = \frac{2\pi}{r} \quad \text{(3)} \]

### Step 4: Substitute \( \theta \) from equation (3) into equation (2).
Substituting \( \theta \) in equation (2):
\[ 6\pi = \frac{1}{2} r^2 \left(\frac{2\pi}{r}\right) \]

### Step 5: Simplify the equation.
This simplifies to:
\[ 6\pi = \frac{1}{2} \cdot 2\pi r \]
\[ 6\pi = \pi r \]

### Step 6: Solve for \( r \).
Dividing both sides by \( \pi \):
\[ 6 = r \]

Thus, the radius \( r \) of the circle is:
\[ \boxed{6} \text{ inches} \]

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