An arc of the `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 length of the arc and the angle subtended by the arc, we can follow these steps:

### Step 1: Understand the relationship between arc length, radius, and angle
The formula for the length of an arc (L) is given by:
\[ L = r \cdot \theta \]
where \( r \) is the radius of the circle and \( \theta \) is the angle in radians.

### Step 2: Convert the angle from degrees to radians
We are given that the angle is 240 degrees. To convert degrees to radians, we use the conversion factor:
\[ \text{Radians} = \text{Degrees} \times \frac{\pi}{180} \]

So, for 240 degrees:
\[ \theta = 240 \times \frac{\pi}{180} = \frac{240\pi}{180} = \frac{4\pi}{3} \text{ radians} \]

### Step 3: Substitute the values into the arc length formula
We know the length of the arc \( L = 16\pi \) units and the angle \( \theta = \frac{4\pi}{3} \) radians. We substitute these values into the arc length formula:
\[ 16\pi = r \cdot \frac{4\pi}{3} \]

### Step 4: Solve for the radius \( r \)
To isolate \( r \), we rearrange the equation:
\[ r = \frac{16\pi}{\frac{4\pi}{3}} \]

This can be simplified:
\[ r = 16\pi \cdot \frac{3}{4\pi} \]

### Step 5: Cancel out \( \pi \) and simplify
The \( \pi \) cancels out:
\[ r = 16 \cdot \frac{3}{4} \]

Now, simplify \( 16 \cdot \frac{3}{4} \):
\[ r = 16 \cdot 0.75 = 12 \text{ units} \]

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

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