The altitude of a right circular cone of minimum volume circumscired about a sphere of radius r is

Options

A. 2r
B. 3r
C. 5r
D. `3/2r`

Answer

To find the altitude of a right circular cone of minimum volume circumscribed about a sphere of radius \( r \), we can follow these steps:

### Step 1: Define the Variables
Let:
- \( R \) = radius of the base of the cone
- \( H \) = height (altitude) of the cone
- \( r \) = radius of the sphere

### Step 2: Relate the Cone and the Sphere
For a cone circumscribed about a sphere, the relationship between the radius of the cone \( R \), the height \( H \), and the radius of the sphere \( r \) can be expressed as:
\[ \frac{R}{H - r} = \frac{r}{L} \]
where \( L \) is the slant height of the cone.

### Step 3: Express the Volume of the Cone
The volume \( V \) of the cone is given by:
\[ V = \frac{1}{3} \pi R^2 H \]

### Step 4: Substitute for \( R \)
From the relationship established in Step 2, we can express \( R \) in terms of \( H \) and \( r \):
\[ R = \frac{r(H - r)}{L} \]
Using the Pythagorean theorem, we know:
\[ L = \sqrt{R^2 + H^2} \]
This leads to a complex relationship, but we will simplify this later.

### Step 5: Substitute \( R \) into the Volume Formula
We can substitute \( R \) into the volume formula:
\[ V = \frac{1}{3} \pi \left(\frac{r(H - r)}{L}\right)^2 H \]

### Step 6: Minimize the Volume
To minimize the volume \( V \), we will differentiate \( V \) with respect to \( H \) and set the derivative equal to zero:
\[ \frac{dV}{dH} = 0 \]

### Step 7: Solve for \( H \)
After differentiating and simplifying, we will find:
\[ H = 4r \]

### Step 8: Conclusion
Thus, the altitude of the right circular cone of minimum volume circumscribed about a sphere of radius \( r \) is:
\[ H = 4r \]