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Question 1181724: 17. Find the volume of the largest right circular cylinder of altitude 8 in. that can be cut from a sphere of diameter 12 in.
Found 2 solutions by CPhill, greenestamps:
Answer by CPhill(1959)   (Show Source):
You can put this solution on YOUR website! 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.**
Answer by greenestamps(13200)  (Show Source):
You can put this solution on YOUR website!
There largest right circular cylinder with height 8 that can be cut from a sphere with diameter 12 is the cylinder that can be inscribed in the sphere.
For that cylinder, each base is 8/2 = 4 units from the center of the sphere. Then, from the Pythagorean Theorem, with the radius of the sphere being 6, the radius of each base of the cylinder is 2*sqrt(5).
And then the volume of the cylinder is
ANSWER: 160pi
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