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Let's analyze the problem. We know that:

1. \( EG \) is an arc of a circle with a length of \( 6\pi \) inches.
2. The angle \( \angle EOF = 30^\circ \), centered at \( O \).
3. We are asked to find the straight-line distance \( EG \) (chord length from \( E \) to \( G \)).

### Step-by-Step Solution

1. **Find the Radius \( r \):**

   The length of an arc \( s \) is given by:
   \[
   s = r \theta
   \]
   where \( \theta \) is the angle in radians. Here, \( s = 6\pi \) and \( \theta = 30^\circ \). First, convert \( 30^\circ \) to radians:
   \[
   \theta = \frac{30^\circ \times \pi}{180^\circ} = \frac{\pi}{6}
   \]
   Now, substitute \( s = 6\pi \) and \( \theta = \frac{\pi}{6} \):
   \[
   6\pi = r \cdot \frac{\pi}{6}
   \]
   Solving for \( r \):
   \[
   r = \frac{6\pi}{\frac{\pi}{6}} = 36
   \]

   So, the radius \( r \) of the circle is 36 inches.

2. **Calculate the Length of \( EG \):**

   In a circle, the length of a chord \( EG \) with a central angle \( \theta \) can be found using:
   \[
   EG = 2r \sin\left(\frac{\theta}{2}\right)
   \]
   Here, \( r = 36 \) and \( \theta = 30^\circ \), so:
   \[
   EG = 2 \cdot 36 \cdot \sin\left(\frac{30^\circ}{2}\right)
   \]
   Simplifying:
   \[
   EG = 2 \cdot 36 \cdot \sin(15^\circ)
   \]

   Using \( \sin(15^\circ) \approx 0.2588 \):
   \[
   EG \approx 2 \cdot 36 \cdot 0.2588 = 18.62
   \]

### Answer
So, \( EG \approx 18.62 \) inches.

Would you like further details on any steps?

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Here are some related questions to explore:

1. How is the formula for arc length derived?
2. What other properties of circles involve radians?
3. How is the chord length formula derived?
4. Why is \( \sin(15^\circ) \) used in this calculation?
5. How can we verify our answer geometrically?

**Tip:** Remember to always convert angles to radians when using arc length or sector area formulas.

Learn how to find the chord length EG when given an arc length of 6π inches and a central angle of 30 degrees. This problem explores concepts in circle geometry, including arc and chord length calculations, and is suitable for high school students.

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