The radius of the circle whose arc of length `15\ pi` cm makes an angle of `(3pi)/4` radians at the centre is

Options

A. 5 cm
B. 10 cm
C. 15 cm
D. 20 cm

Answer

To find the radius of the circle given the arc length and the angle in radians, we can use the formula:

\[ \text{Angle in radians} = \frac{\text{Length of arc}}{\text{Radius}} \]

Given:
- Length of arc \( L = 15\pi \) cm
- Angle \( \theta = \frac{3\pi}{4} \) radians

We need to find the radius \( r \).

### Step 1: Write the formula
The formula relating arc length, radius, and angle in radians is:

\[ \theta = \frac{L}{r} \]

### Step 2: Substitute the known values
Substituting the known values into the formula:

\[ \frac{3\pi}{4} = \frac{15\pi}{r} \]

### Step 3: Cross-multiply to solve for \( r \)
Cross-multiplying gives:

\[ 3\pi \cdot r = 15\pi \cdot 4 \]

### Step 4: Simplify the equation
This simplifies to:

\[ 3\pi r = 60\pi \]

### Step 5: Divide both sides by \( 3\pi \)
To isolate \( r \), divide both sides by \( 3\pi \):

\[ r = \frac{60\pi}{3\pi} \]

### Step 6: Simplify the fraction
The \( \pi \) cancels out:

\[ r = \frac{60}{3} = 20 \]

### Conclusion
Thus, the radius of the circle is:

\[ \boxed{20 \text{ cm}} \]