We are given a chord of 25 mm in a circle, and the angle subtended by this chord at the center is $\frac{\pi}{6}$ radians. We need to find the length of the arc cut off by the chord to one decimal place.

Step-by-Step Solution:

1. Identify the central angle ($\theta$) and radius of the circle:

   • The angle subtended at the center is $\theta = \frac{\pi}{6}$ radians.
   • To calculate the length of the arc, we need the radius of the circle, which is currently unknown.

2. Relate chord length to the radius:

   • The formula for the length of a chord in a circle is given by: $\text{Chord length} = 2r \sin\left(\frac{\theta}{2}\right)$ where $r$ is the radius and $\theta$ is the central angle.
   • Substituting the values we know: $25 = 2r \sin\left(\frac{\pi}{12}\right)$ Solving for $r$: $r = \frac{25}{2 \sin\left(\frac{\pi}{12}\right)}$

3. Calculate the value of $r$: Let me compute this first.The radius of the circle is approximately $48.3 \, \text{mm}$.

4. Calculate the length of the arc: The formula for the length of an arc is given by: $\text{Arc length} = r \theta$ Substituting the values $r = 48.3 \, \text{mm}$ and $\theta = \frac{\pi}{6}$: $\text{Arc length} = 48.3 \times \frac{\pi}{6}$ Let me calculate that.It seems like I can’t do more advanced data analysis right now. Please try again later.

Would you like me to guide you through the manual calculation for the arc length, or do you have any other questions?

Here are five related questions that you might find interesting:

1. How can you find the area of a sector given the radius and central angle?
2. What is the relationship between a chord and a circle's radius?
3. How do you calculate the circumference of a circle if only the chord length is known?
4. How does the subtended angle at the center relate to the arc's length?
5. How can you find the radius of a circle given the length of an arc and the angle?

Tip: Remember, for radians, the arc length formula simplifies to $r \cdot \theta$, where $\theta$ is in radians.