A cone is inscribed in a sphere of radius 12 cm. If the volume of the cone is maximum, find its height.

Answer

To find the height of the cone inscribed in a sphere of radius 12 cm that maximizes the volume, we can follow these steps:

### Step 1: Understand the Geometry
We have a sphere of radius \( R = 12 \) cm and a cone inscribed in it. Let the height of the cone be \( h \) and the radius of the base of the cone be \( r \).

### Step 2: Relate the Cone's Dimensions to the Sphere
The cone's vertex is at the bottom of the sphere, and the center of the sphere is at the midpoint of the height of the cone. If we let \( x \) be the distance from the center of the sphere to the base of the cone, then we can express the height of the cone as:
\[ h = 12 + x \]
Using the Pythagorean theorem in the right triangle formed by the radius of the sphere, the radius of the cone, and the height from the center of the sphere to the base of the cone, we have:
\[ R^2 = r^2 + x^2 \]
Substituting \( R = 12 \):
\[ 12^2 = r^2 + x^2 \implies 144 = r^2 + x^2 \implies r^2 = 144 - x^2 \]

### Step 3: Write the Volume of the Cone
The volume \( V \) of the cone is given by:
\[ V = \frac{1}{3} \pi r^2 h \]
Substituting \( r^2 \) and \( h \):
\[ V = \frac{1}{3} \pi (144 - x^2)(12 + x) \]

### Step 4: Simplify the Volume Expression
Expanding the volume expression:
\[ V = \frac{1}{3} \pi (144 \cdot 12 + 144x - 12x^2 - x^3) \]
\[ V = \frac{1}{3} \pi (1728 + 144x - 12x^2 - x^3) \]

### Step 5: Differentiate the Volume Function
To find the maximum volume, we differentiate \( V \) with respect to \( x \):
\[ \frac{dV}{dx} = \frac{1}{3} \pi (144 - 24x - 3x^2) \]

### Step 6: Set the Derivative to Zero
Setting the derivative equal to zero to find critical points:
\[ 144 - 24x - 3x^2 = 0 \]
This simplifies to:
\[ 3x^2 + 24x - 144 = 0 \]
Dividing through by 3:
\[ x^2 + 8x - 48 = 0 \]

### Step 7: Solve the Quadratic Equation
Using the quadratic formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \):
\[ x = \frac{-8 \pm \sqrt{8^2 - 4 \cdot 1 \cdot (-48)}}{2 \cdot 1} = \frac{-8 \pm \sqrt{64 + 192}}{2} = \frac{-8 \pm \sqrt{256}}{2} = \frac{-8 \pm 16}{2} \]
Calculating the roots:
\[ x = \frac{8}{2} = 4 \quad \text{or} \quad x = \frac{-24}{2} = -12 \]
Since height cannot be negative, we take \( x = 4 \).

### Step 8: Find the Height of the Cone
Substituting \( x = 4 \) back to find the height:
\[ h = 12 + x = 12 + 4 = 16 \text{ cm} \]

### Final Answer
The height of the cone when its volume is maximum is \( \boxed{16} \) cm.