Find the maximum volume of the cylinder which can be inscribed in a sphere of radius `3sqrt(3)cm` (Leave the answer in terms of `pi`)

Answer

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

### Step 1: Write the formula for the volume of a cylinder
The volume \(V\) of a cylinder is given by the formula:
\[ V = \pi r^2 h \]
where \(r\) is the radius of the cylinder and \(h\) is its height.

### Step 2: Relate the dimensions of the cylinder to the sphere
In a sphere of radius \(R\), the relationship between the radius of the cylinder \(r\) and its height \(h\) can be derived using the Pythagorean theorem. The radius of the sphere is \(R = 3\sqrt{3}\) cm, and we can write:
\[ R^2 = r^2 + \left(\frac{h}{2}\right)^2 \]
Substituting \(R\):
\[ (3\sqrt{3})^2 = r^2 + \left(\frac{h}{2}\right)^2 \]
This simplifies to:
\[ 27 = r^2 + \frac{h^2}{4} \]

### Step 3: Express \(r^2\) in terms of \(h\)
Rearranging the equation gives:
\[ r^2 = 27 - \frac{h^2}{4} \]

### Step 4: Substitute \(r^2\) into the volume formula
Now substitute \(r^2\) into the volume formula:
\[ V = \pi \left(27 - \frac{h^2}{4}\right) h \]
This expands to:
\[ V = 27\pi h - \frac{\pi h^3}{4} \]

### Step 5: Differentiate the volume with respect to height \(h\)
To find the maximum volume, we differentiate \(V\) with respect to \(h\):
\[ \frac{dV}{dh} = 27\pi - \frac{3\pi h^2}{4} \]

### Step 6: Set the derivative to zero to find critical points
Setting the derivative equal to zero:
\[ 27\pi - \frac{3\pi h^2}{4} = 0 \]
Solving for \(h^2\):
\[ 27\pi = \frac{3\pi h^2}{4} \]
\[ h^2 = 36 \quad \Rightarrow \quad h = 6 \text{ cm} \quad (\text{since height must be positive}) \]

### Step 7: Find the radius \(r\)
Now substitute \(h = 6\) cm back into the equation for \(r^2\):
\[ r^2 = 27 - \frac{6^2}{4} = 27 - 9 = 18 \]
Thus,
\[ r = \sqrt{18} = 3\sqrt{2} \text{ cm} \]

### Step 8: Calculate the maximum volume
Now substitute \(r\) and \(h\) back into the volume formula:
\[ V = \pi r^2 h = \pi (18)(6) = 108\pi \text{ cm}^3 \]

### Final Answer
The maximum volume of the cylinder that can be inscribed in the sphere is:
\[ \boxed{108\pi \text{ cm}^3} \]