Question 1181724
Here's how to find the volume of the largest cylinder:

**1. Visualize the Setup:** Imagine a sphere with a cylinder inside it. The cylinder's height is fixed at 8 inches, and we want to find the largest possible radius for the cylinder.

**2. Key Dimensions:**

*   Sphere diameter = 12 inches, so sphere radius (R) = 6 inches.
*   Cylinder height (h) = 8 inches.
*   Let 'r' be the radius of the cylinder.

**3. Cross-Section:**  A cross-section through the center of the sphere and cylinder reveals a circle (the sphere) with a rectangle (the cylinder) inside.  The diagonal of this rectangle is the diameter of the sphere (12 inches).

**4. Pythagorean Theorem:** We can use the Pythagorean theorem to relate the sphere's radius (R), the cylinder's radius (r), and *half* of the cylinder's height (h/2):

R² = r² + (h/2)²

**5. Solve for the Cylinder's Radius (r):**

6² = r² + (8/2)²
36 = r² + 16
r² = 20
r = √20 = 2√5 inches

**6. Volume of the Cylinder:**

Volume of a cylinder = πr²h
V = π(2√5)² * 8
V = π * 20 * 8
V = 160π cubic inches

**Therefore, the volume of the largest right circular cylinder is 160π cubic inches, or approximately 502.65 cubic inches.**