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This problem involves a circle, where \( AC \) is an arc with a given length of \( 4\pi \) inches, and the task is to find the length of \( AD \). Let's break it down step-by-step.

### Step 1: Analyze the Circle and Arc
- \( AC \) is an arc of the circle, and its length is given as \( 4\pi \) inches.
- \( \angle AOC \) is subtended by the arc \( AC \) at the center of the circle, with the measure of \( 30^\circ \) given.

### Step 2: Relationship between Arc Length and Angle
The length of an arc \( s \) in a circle is given by the formula:
\[
s = r \theta
\]
where:
- \( r \) is the radius of the circle,
- \( \theta \) is the angle subtended by the arc at the center (in radians).

Here, the arc length \( s = 4\pi \) inches and the angle \( \theta = 30^\circ \). We need to convert the angle into radians:
\[
\theta = \frac{30^\circ \times \pi}{180^\circ} = \frac{\pi}{6} \, \text{radians}.
\]

### Step 3: Solve for the Radius \( r \)
Now, substitute the values into the arc length formula:
\[
4\pi = r \times \frac{\pi}{6}.
\]
Simplify:
\[
r = \frac{4\pi}{\pi/6} = 24 \, \text{inches}.
\]
So, the radius of the circle is \( r = 24 \) inches.

### Step 4: Find \( AD \)
The line segment \( AD \) is half of the diameter of the circle because it passes through the center \( O \) of the circle. Since the radius \( r = 24 \) inches, the full diameter \( AB \) is \( 2r = 48 \) inches. Therefore:
\[
AD = \frac{AB}{2} = \frac{48}{2} = 24 \, \text{inches}.
\]

Thus, \( AD = 24 \) inches.

Let me know if you'd like further clarification or have any questions.

### Follow-up Questions:
1. How do you convert degrees to radians?
2. What is the formula for the length of an arc in a circle?
3. How is the diameter of a circle related to its radius?
4. How do you calculate the radius when given the arc length and angle?
5. What is the central angle in radians for different degrees?

**Tip:** Always remember to convert angles to radians when using formulas involving arc lengths or angular velocity in a circle.

Solve a geometry problem involving the arc length of a circle and central angles. Learn how to find AD by calculating the radius of the circle and applying arc length formulas. This problem is suitable for Grades 10-12.

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