A wire of length 121 cm is bent so as to lie along the arc of a circle of radius 180 cm. Find in degrees, the angle subtended at the centre by the arc.

Answer

To find the angle subtended at the center of the circle by the arc formed by the wire, we can use the formula that relates the length of the arc (L), the radius of the circle (r), and the angle in radians (θ):

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

### Step-by-step Solution:

1. **Identify the given values:**
- Length of the wire (arc length), \( L = 121 \, \text{cm} \)
- Radius of the circle, \( r = 180 \, \text{cm} \)

2. **Substitute the values into the formula:**
\[ \theta = \frac{L}{r} = \frac{121}{180} \]

3. **Calculate θ in radians:**
\[ \theta = \frac{121}{180} \, \text{radians} \]

4. **Convert θ from radians to degrees:**
To convert radians to degrees, use the conversion factor \( \frac{180}{\pi} \):
\[ \theta_{\text{degrees}} = \theta \times \frac{180}{\pi} = \frac{121}{180} \times \frac{180}{\pi} \]
Simplifying this gives:
\[ \theta_{\text{degrees}} = \frac{121}{\pi} \approx 38.54 \, \text{degrees} \]

5. **Final Answer:**
The angle subtended at the center by the arc is approximately \( 38.54 \, \text{degrees} \).