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This problem involves finding the volume of a "bead," which is created by removing a cylindrical portion from the center of a sphere. To find the volume, we'll need to compute the volume of the sphere and subtract the volume of the cylindrical section.

### Step 1: Volume of the Sphere
The volume of a sphere is given by the formula:
\[
V_{\text{sphere}} = \frac{4}{3} \pi R^3
\]
Where \( R \) is the radius of the sphere.

### Step 2: Volume of the Cylinder
The cylindrical section removed from the sphere has a radius \( r \) and height \( h \). The volume of a cylinder is:
\[
V_{\text{cylinder}} = \pi r^2 h
\]
However, \( h \), the height of the cylindrical section, is equal to the diameter of the sphere, which is \( 2R \).

So, the volume of the cylinder becomes:
\[
V_{\text{cylinder}} = \pi r^2 \cdot 2R = 2\pi r^2 R
\]

### Step 3: Volume of the Bead
To find the volume of the bead, we subtract the volume of the cylinder from the volume of the sphere:
\[
V_{\text{bead}} = V_{\text{sphere}} - V_{\text{cylinder}}
\]

Substituting the formulas:
\[
V_{\text{bead}} = \frac{4}{3} \pi R^3 - 2 \pi r^2 R
\]

### Step 4: Plug in the Given Values
We are given \( R = 6 \) and \( r = 4 \).

Substitute these values into the formula:
\[
V_{\text{bead}} = \frac{4}{3} \pi (6)^3 - 2 \pi (4)^2 (6)
\]

First, calculate each part:
- \( \frac{4}{3} \pi (6)^3 = \frac{4}{3} \pi (216) = 288 \pi \)
- \( 2 \pi (4)^2 (6) = 2 \pi (16)(6) = 192 \pi \)

Thus:
\[
V_{\text{bead}} = 288 \pi - 192 \pi = 96 \pi
\]

### Final Answer:
The volume of the bead is:
\[
V_{\text{bead}} = 96 \pi \, \text{cubic units}
\]

Would you like further clarification or details on any step?

Here are five related questions to consider:
1. How would the volume change if the radius \( r \) of the cylinder were increased?
2. What would happen to the bead's volume if the sphere's radius \( R \) were halved?
3. Can you express the volume of the bead in terms of general variables \( R \) and \( r \)?
4. How does the volume of the removed cylinder compare to the total volume of the sphere?
5. What would the formula be for the surface area of the bead?

**Tip:** In problems involving volume subtraction, it's essential to ensure that the volumes being subtracted are for matching geometric regions (such as both being cylinders, or both spherical regions).

Find the volume of the bead formed by removing a cylinder of radius r = 4 from the center of a sphere of radius R = 6.

Learn how to find the volume of a bead by removing a cylindrical section from the center of a sphere with radius 6. This problem involves key geometric volume formulas, including the volumes of spheres and cylinders.

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