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.