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

Options

A. `30^(@)`
B. `36.27^(@)`
C. `45^(@)`
D. `52^(@)`

Answer

To find the angle \( \theta \) subtended by an arc of length 19 cm in a circle of radius 30 cm, we can use the formula for the length of an arc:

\[ L = \frac{\pi r \theta}{180} \]

where:
- \( L \) is the length of the arc,
- \( r \) is the radius of the circle,
- \( \theta \) is the angle in degrees.

### Step 1: Substitute the known values into the formula.

Given:
- Length of the arc \( L = 19 \) cm
- Radius \( r = 30 \) cm

Substituting these values into the formula:

\[ 19 = \frac{\pi \times 30 \times \theta}{180} \]

### Step 2: Rearrange the equation to solve for \( \theta \).

To isolate \( \theta \), we can multiply both sides by 180:

\[ 19 \times 180 = \pi \times 30 \times \theta \]

This simplifies to:

\[ 3420 = \pi \times 30 \times \theta \]

Now, divide both sides by \( \pi \times 30 \):

\[ \theta = \frac{3420}{\pi \times 30} \]

### Step 3: Calculate \( \theta \).

Using \( \pi \approx \frac{22}{7} \):

\[ \theta = \frac{3420}{\frac{22}{7} \times 30} \]

Calculating the denominator:

\[ \frac{22}{7} \times 30 = \frac{660}{7} \]

Now substituting back into the equation for \( \theta \):

\[ \theta = \frac{3420 \times 7}{660} \]

Calculating \( \frac{3420 \times 7}{660} \):

\[ \theta = \frac{23940}{660} = 36.27 \]

### Final Answer:

Thus, the value of \( \theta \) is approximately \( 36.27^\circ \).