An arc of length of length 19 cm of a circle of radius 30 cm, subtends an angle `theta` at the centre O. The value of `theta` is:

Options

A. `30^(@)`
B. `37^(@)`
C. `45^(@)`
D. `52^(@)`

Answer

To find the angle \(\theta\) subtended at the center of a circle by an arc of length 19 cm, we can use the formula for the length of an arc:

\[ L = \frac{\theta}{360} \times 2\pi r \]

Where:
- \(L\) 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**:
- Length of the arc, \(L = 19 \, \text{cm}\)
- Radius of the circle, \(r = 30 \, \text{cm}\)

2. **Substitute the values into the arc length formula**:
\[ 19 = \frac{\theta}{360} \times 2\pi \times 30 \]

3. **Calculate \(2\pi r\)**:
- Using \(\pi \approx \frac{22}{7}\):
\[ 2\pi \times 30 = 2 \times \frac{22}{7} \times 30 = \frac{1320}{7} \]

4. **Rewrite the equation**:
\[ 19 = \frac{\theta}{360} \times \frac{1320}{7} \]

5. **Multiply both sides by \(360\)** to eliminate the fraction:
\[ 19 \times 360 = \frac{\theta \times 1320}{7} \]

6. **Calculate \(19 \times 360\)**:
\[ 19 \times 360 = 6840 \]

7. **Multiply both sides by \(7\)** to get rid of the denominator:
\[ 6840 \times 7 = 1320\theta \]
\[ 47880 = 1320\theta \]

8. **Solve for \(\theta\)**:
\[ \theta = \frac{47880}{1320} \]

9. **Perform the division**:
\[ \theta = 36.27 \]

10. **Round off to the nearest whole number**:
\[ \theta \approx 37^\circ \]

### Final Answer:
\(\theta \approx 37^\circ\)