An arc length of `16pi` units subtends an angle of 240 degrees. Find the radius (in units) of the circle.

Options

A. 6
B. 12
C. 24
D. 36

Answer

To find the radius of the circle given the arc length and the angle subtended by the arc, we can use the formula for the length of an arc:

\[ \text{Arc Length} = \frac{\theta}{360} \times 2\pi r \]

where:
- \(\text{Arc Length}\) is the length of the arc,
- \(\theta\) is the angle in degrees,
- \(r\) is the radius of the circle.

### Step-by-Step Solution:

1. **Identify the given values:**
- Arc Length = \(16\pi\) units
- Angle \(\theta = 240\) degrees

2. **Substitute the values into the arc length formula:**
\[ 16\pi = \frac{240}{360} \times 2\pi r \]

3. **Simplify the fraction \(\frac{240}{360}\):**
\[ \frac{240}{360} = \frac{2}{3} \]
So, the equation becomes:
\[ 16\pi = \frac{2}{3} \times 2\pi r \]

4. **Multiply \(\frac{2}{3}\) by \(2\pi\):**
\[ 16\pi = \frac{4\pi}{3} r \]

5. **To eliminate \(\pi\) from both sides, divide by \(\pi\):**
\[ 16 = \frac{4}{3} r \]

6. **Multiply both sides by \(3\) to eliminate the fraction:**
\[ 48 = 4r \]

7. **Divide both sides by \(4\) to solve for \(r\):**
\[ r = \frac{48}{4} = 12 \]

8. **Conclusion:**
The radius of the circle is \(12\) units.

### Final Answer:
The radius of the circle is \(12\) units.

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