A cube of maximum possible volume is cut from the solid right circular cylinder. What is the ratio of volume of cube to that of cylinder if the edge of a cube is equal to the height of the cylinder?

Options

A. `11/7`
B. `sqrt2pi/7`
C. `7/11`
D. none of these

Answer

To solve the problem, we need to find the ratio of the volume of a cube to the volume of a cylinder, given that the edge of the cube is equal to the height of the cylinder.

### Step-by-Step Solution:

1. **Define Variables**:
Let the edge of the cube be \( a \). According to the problem, the height of the cylinder \( h \) is also equal to \( a \).

2. **Volume of the Cube**:
The volume \( V_c \) of a cube is given by the formula:
\[ V_c = a^3 \]

3. **Volume of the Cylinder**:
The volume \( V_{cy} \) of a cylinder is given by the formula:
\[ V_{cy} = \pi r^2 h \]
Since the edge of the cube is equal to the height of the cylinder, we can substitute \( h \) with \( a \):
\[ V_{cy} = \pi r^2 a \]

4. **Finding the Radius**:
For the cube to fit inside the cylinder, the diagonal of the cube's base must be equal to the diameter of the cylinder. The diagonal \( d \) of the base of the cube (which is a square) can be calculated using the formula:
\[ d = a\sqrt{2} \]
Since the diameter of the cylinder is \( 2r \), we set the diagonal equal to the diameter:
\[ a\sqrt{2} = 2r \]
From this, we can express \( r \) in terms of \( a \):
\[ r = \frac{a\sqrt{2}}{2} \]

5. **Substituting \( r \) into the Volume of the Cylinder**:
Now, substitute \( r \) back into the volume formula of the cylinder:
\[ V_{cy} = \pi \left(\frac{a\sqrt{2}}{2}\right)^2 a \]
Simplifying this:
\[ V_{cy} = \pi \left(\frac{2a^2}{4}\right) a = \pi \left(\frac{a^2}{2}\right) a = \frac{\pi a^3}{2} \]

6. **Finding the Ratio of Volumes**:
Now, we can find the ratio of the volume of the cube to the volume of the cylinder:
\[ \text{Ratio} = \frac{V_c}{V_{cy}} = \frac{a^3}{\frac{\pi a^3}{2}} = \frac{2}{\pi} \]

### Final Answer:
The ratio of the volume of the cube to that of the cylinder is:
\[ \frac{2}{\pi} \]