Find the height of right circular cylinder of maximum volume that can be inscribed in a sphere of radius `10sqrt3cm`.

Answer

To find the height of the right circular cylinder of maximum volume that can be inscribed in a sphere of radius \(10\sqrt{3}\) cm, we can follow these steps:

### Step 1: Understand the Geometry
We have a sphere with radius \(R = 10\sqrt{3}\) cm. We want to inscribe a cylinder in this sphere. Let the height of the cylinder be \(h\) and the radius of the cylinder be \(r\).

### Step 2: Relate the Cylinder and Sphere
The cylinder is inscribed in the sphere, so we can use the Pythagorean theorem. The height of the cylinder can be divided into two equal parts from the center of the sphere to the top and bottom of the cylinder. Let \(x\) be the distance from the center of the sphere to the top (or bottom) of the cylinder. Then, the height of the cylinder is:
\[ h = 2x \]
The radius of the cylinder is related to \(x\) and \(R\) by:
\[ r^2 + x^2 = R^2 \]
Thus,
\[ r^2 = R^2 - x^2 \]

### Step 3: Write the Volume of the Cylinder
The volume \(V\) of the cylinder is given by:
\[ V = \pi r^2 h = \pi (R^2 - x^2)(2x) \]
Substituting \(R = 10\sqrt{3}\):
\[ V = \pi ((10\sqrt{3})^2 - x^2)(2x) = \pi (300 - x^2)(2x) = 2\pi (300x - x^3) \]

### Step 4: Differentiate the Volume
To find the maximum volume, we need to differentiate \(V\) with respect to \(x\):
\[ \frac{dV}{dx} = 2\pi (300 - 3x^2) \]
Setting the derivative equal to zero to find critical points:
\[ 300 - 3x^2 = 0 \implies 3x^2 = 300 \implies x^2 = 100 \implies x = 10 \]

### Step 5: Determine the Height
Now, we can find the height \(h\) of the cylinder:
\[ h = 2x = 2 \times 10 = 20 \text{ cm} \]

### Step 6: Verify Maximum Volume
To ensure that this value of \(x\) gives a maximum volume, we can check the second derivative:
\[ \frac{d^2V}{dx^2} = -6\pi x \]
Since \(x = 10\) is positive, \(\frac{d^2V}{dx^2} < 0\), confirming that we have a maximum.

### Final Answer
Thus, the height of the right circular cylinder of maximum volume that can be inscribed in a sphere of radius \(10\sqrt{3}\) cm is:
\[ \boxed{20 \text{ cm}} \]