The problem states:

A wire $121\ \text{cm}$ long is bent so as to lie along the arc of a circle of radius $180\ \text{cm}$. Find the angle subtended at the center by the wire.

Step-by-Step Solution:

1. Concepts Involved:

   • The length of an arc ($l$) of a circle is related to the radius ($r$) and the central angle ($\theta$) in radians by the formula: $l = r \theta$
   • Here, $l = 121\ \text{cm}$ and $r = 180\ \text{cm}$, and we need to find $\theta$.

2. Rearranging the Formula: Solve for $\theta$: $\theta = \frac{l}{r}$

3. Substitute the Values: $\theta = \frac{121}{180}$

4. Simplify the Fraction: Perform the division: $\theta = 0.6722\ \text{radians} \ (\text{approx. to 4 decimal places})$

5. Convert to Degrees (if needed): To convert radians to degrees, use the formula: $\text{Angle in degrees} = \theta \times \frac{180}{\pi}$ Substituting: $\text{Angle in degrees} = 0.6722 \times \frac{180}{\pi} \approx 38.52^\circ$

Final Answer:

• In radians: $\theta \approx 0.6722\ \text{radians}$
• In degrees: $\theta \approx 38.52^\circ$

Would you like further clarification or assistance? Here are some related questions:

1. What are the steps to convert between radians and degrees?
2. How is the formula $l = r\theta$ derived?
3. Why is the angle subtended in radians often more useful in mathematical contexts?
4. Can this formula be used for arcs of ellipses or other curves?
5. What would happen to the angle if the radius were doubled?

Tip: Always remember to double-check units (radians vs. degrees) to avoid confusion in your final answer!