The radius of a cone is √2 times the height of the cone. A cube of maximum possible volume is cut from the same cone. What is the ratio of the volume of the cone to the volume of the cube?

Options

A. `3.18 pi`
B. `2. 25 pi`
C. `2.35 pi`
D. can't be determined

Answer

To solve the problem, we need to find the ratio of the volume of a cone to the volume of a cube that can be inscribed within the cone. Let's break down the steps:

### Step 1: Define the Variables
Let the height of the cone be \( h \). According to the problem, the radius \( r \) of the cone is given by:
\[ r = \sqrt{2} \times h \]

### Step 2: Determine the Volume of the Cone
The volume \( V \) of a cone is given by the formula:
\[ V = \frac{1}{3} \pi r^2 h \]
Substituting the expression for \( r \):
\[ V = \frac{1}{3} \pi (\sqrt{2} h)^2 h = \frac{1}{3} \pi (2h^2) h = \frac{2}{3} \pi h^3 \]

### Step 3: Determine the Side Length of the Cube
Let the side length of the cube be \( a \). The cube is inscribed in the cone, meaning its corners touch the sides of the cone. The height of the cone above the cube is \( h - a \).

Using the properties of similar triangles, we can set up the following ratio:
\[ \frac{a}{\frac{r}{2}} = \frac{h - a}{h} \]
Where \( \frac{r}{2} \) is the radius of the cone at the height where the cube is inscribed. Since \( r = \sqrt{2} h \):
\[ \frac{a}{\frac{\sqrt{2} h}{2}} = \frac{h - a}{h} \]
This simplifies to:
\[ \frac{a}{\frac{\sqrt{2}}{2} h} = \frac{h - a}{h} \]
Cross-multiplying gives:
\[ a h = \frac{\sqrt{2}}{2} h (h - a) \]

### Step 4: Solve for \( a \)
Rearranging the equation:
\[ a h = \frac{\sqrt{2}}{2} h^2 - \frac{\sqrt{2}}{2} a h \]
Combine like terms:
\[ a h + \frac{\sqrt{2}}{2} a h = \frac{\sqrt{2}}{2} h^2 \]
Factoring out \( a \):
\[ a \left( h + \frac{\sqrt{2}}{2} h \right) = \frac{\sqrt{2}}{2} h^2 \]
Thus:
\[ a = \frac{\frac{\sqrt{2}}{2} h^2}{h \left( 1 + \frac{\sqrt{2}}{2} \right)} = \frac{\frac{\sqrt{2}}{2} h}{1 + \frac{\sqrt{2}}{2}} \]
Simplifying gives:
\[ a = \frac{\sqrt{2} h}{2 + \sqrt{2}} \]

### Step 5: Calculate the Volume of the Cube
The volume \( V_{cube} \) of the cube is:
\[ V_{cube} = a^3 = \left( \frac{\sqrt{2} h}{2 + \sqrt{2}} \right)^3 \]

### Step 6: Find the Ratio of Volumes
Now we need to find the ratio of the volume of the cone to the volume of the cube:
\[ \text{Ratio} = \frac{V_{cone}}{V_{cube}} = \frac{\frac{2}{3} \pi h^3}{\left( \frac{\sqrt{2} h}{2 + \sqrt{2}} \right)^3} \]
Calculating the volume of the cube:
\[ V_{cube} = \frac{2h^3}{(2 + \sqrt{2})^3} \]
Thus, the ratio becomes:
\[ \text{Ratio} = \frac{\frac{2}{3} \pi h^3}{\frac{2h^3}{(2 + \sqrt{2})^3}} = \frac{\frac{2}{3} \pi (2 + \sqrt{2})^3}{2} \]
This simplifies to:
\[ \text{Ratio} = \frac{\pi (2 + \sqrt{2})^3}{3} \]

### Final Calculation
Calculating \( (2 + \sqrt{2})^3 \) and simplifying will yield the final ratio.