A cylinder is inscribed in a sphere of radius r
What is the radius of the cylinder of maximum volume ?

Options

A. `(2r)/sqrt(3)`
B. `sqrt(2)/(sqrt(3)``r`
C. r
D. `sqrt(3r)`

Explanation

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

### Step 1: Understand the Geometry
We have a cylinder inscribed in a sphere. The radius of the sphere is \( r \), and we denote the radius of the cylinder as \( R \) and its height as \( h \).

### Step 2: Relate the Cylinder and Sphere
Using the geometry of the situation, we can apply the Pythagorean theorem. The relationship can be expressed as:
\[ R^2 + \left(\frac{h}{2}\right)^2 = r^2 \]
This equation arises because the radius of the sphere is the hypotenuse of a right triangle formed by the radius of the cylinder and half the height of the cylinder.

### Step 3: Express Height in Terms of Radius
From the Pythagorean theorem, we can express \( h \) in terms of \( R \):
\[ h = 2\sqrt{r^2 - R^2} \]

### Step 4: Write the Volume of the Cylinder
The volume \( V \) of the cylinder is given by:
\[ V = \pi R^2 h \]
Substituting \( h \) from the previous step:
\[ V = \pi R^2 \cdot 2\sqrt{r^2 - R^2} = 2\pi R^2 \sqrt{r^2 - R^2} \]

### Step 5: Differentiate to Find Maximum Volume
To find the maximum volume, we need to differentiate \( V \) with respect to \( R \) and set the derivative equal to zero:
\[ \frac{dV}{dR} = 2\pi \left( 2R \sqrt{r^2 - R^2} + R^2 \cdot \frac{-R}{\sqrt{r^2 - R^2}} \right) = 0 \]
This simplifies to:
\[ 2R \sqrt{r^2 - R^2} - \frac{R^3}{\sqrt{r^2 - R^2}} = 0 \]
Multiplying through by \( \sqrt{r^2 - R^2} \):
\[ 2R(r^2 - R^2) - R^3 = 0 \]
\[ 2Rr^2 - 3R^3 = 0 \]
Factoring out \( R \):
\[ R(2r^2 - 3R^2) = 0 \]
Thus, we have two cases: \( R = 0 \) or \( 2r^2 - 3R^2 = 0 \).

### Step 6: Solve for \( R \)
From \( 2r^2 - 3R^2 = 0 \):
\[ 3R^2 = 2r^2 \implies R^2 = \frac{2r^2}{3} \implies R = \sqrt{\frac{2}{3}} r \]

### Conclusion
The radius of the cylinder of maximum volume inscribed in a sphere of radius \( r \) is:
\[ R = \frac{r\sqrt{2}}{\sqrt{3}} \]