Find the volume of the larges cylinder that can be inscribed in a sphere of radius `r`

Answer

To find the volume of the largest cylinder that can be inscribed in a sphere of radius \( r \), we can follow these steps:

### Step 1: Understand the Geometry
We have a sphere with radius \( r \) and a cylinder inscribed within it. Let the radius of the cylinder be \( R \) and the height of the cylinder be \( H \). The center of the sphere and the cylinder coincide.

### Step 2: Relate the Cylinder and Sphere Dimensions
Using the right triangle formed by the radius of the sphere, the height of the cylinder, and the radius of the cylinder, we can set up the following relationship:
- The height of the cylinder is \( H \), thus the distance from the center of the sphere to the top of the cylinder is \( \frac{H}{2} \).
- The radius of the cylinder is \( R \).

Using the Pythagorean theorem in the triangle formed:
\[ r^2 = R^2 + \left(\frac{H}{2}\right)^2 \]
This can be rearranged to express \( R^2 \) in terms of \( H \):
\[ R^2 = r^2 - \left(\frac{H}{2}\right)^2 \]
\[ R^2 = r^2 - \frac{H^2}{4} \]

### Step 3: Write the Volume of the Cylinder
The volume \( V \) of the cylinder can be expressed as:
\[ V = \pi R^2 H \]
Substituting \( R^2 \) from the previous step:
\[ V = \pi \left(r^2 - \frac{H^2}{4}\right) H \]
\[ V = \pi \left(r^2 H - \frac{H^3}{4}\right) \]

### Step 4: Differentiate the Volume with Respect to Height
To find the maximum volume, we differentiate \( V \) with respect to \( H \):
\[ \frac{dV}{dH} = \pi \left(r^2 - \frac{3H^2}{4}\right) \]
Setting the derivative equal to zero to find critical points:
\[ r^2 - \frac{3H^2}{4} = 0 \]
\[ \frac{3H^2}{4} = r^2 \]
\[ H^2 = \frac{4r^2}{3} \]
\[ H = \frac{2r}{\sqrt{3}} \]

### Step 5: Find the Radius of the Cylinder
Substituting \( H \) back into the equation for \( R^2 \):
\[ R^2 = r^2 - \frac{H^2}{4} \]
\[ R^2 = r^2 - \frac{1}{4} \left(\frac{4r^2}{3}\right) \]
\[ R^2 = r^2 - \frac{r^2}{3} \]
\[ R^2 = \frac{2r^2}{3} \]
\[ R = \frac{r\sqrt{2}}{\sqrt{3}} \]

### Step 6: Calculate Maximum Volume
Now substituting \( R \) and \( H \) back into the volume formula:
\[ V = \pi R^2 H \]
\[ V = \pi \left(\frac{2r^2}{3}\right) \left(\frac{2r}{\sqrt{3}}\right) \]
\[ V = \pi \cdot \frac{4r^3}{3\sqrt{3}} \]

### Final Answer
Thus, the volume of the largest cylinder that can be inscribed in a sphere of radius \( r \) is:
\[ V = \frac{4\pi r^3}{3\sqrt{3}} \]