Let the largest possible right circular cone and largest possible sphere be fitted into two cubes of same length. Let C and S denote the volume of cone and volume of sphere respectively, then which one of the following is correct?

Options

A. C = 2S
B. S=2C
C. C=S
D. C = 3S

Answer

To solve the problem, we need to find the volumes of the largest possible right circular cone and the largest possible sphere that can fit into cubes of the same side length, denoted as \( A \).

### Step-by-Step Solution:

1. **Identify the Side Length of the Cube:**
Let the side length of the cube be \( A \).

2. **Determine the Radius and Height of the Cone:**
- The largest right circular cone that can fit into the cube will have:
- Radius \( r = \frac{A}{2} \) (half of the side length)
- Height \( h = A \) (equal to the side length of the cube)

3. **Calculate the Volume of the Cone:**
The formula for the volume \( V \) of a cone is given by:
\[ V = \frac{1}{3} \pi r^2 h \]
Substituting the values of \( r \) and \( h \):
\[ V_C = \frac{1}{3} \pi \left(\frac{A}{2}\right)^2 A \]
Simplifying this:
\[ V_C = \frac{1}{3} \pi \left(\frac{A^2}{4}\right) A = \frac{1}{3} \pi \frac{A^3}{4} = \frac{\pi A^3}{12} \]

4. **Determine the Radius of the Sphere:**
- The largest sphere that can fit into the cube will have:
- Radius \( R = \frac{A}{2} \) (half of the side length)

5. **Calculate the Volume of the Sphere:**
The formula for the volume \( V \) of a sphere is given by:
\[ V = \frac{4}{3} \pi R^3 \]
Substituting the value of \( R \):
\[ V_S = \frac{4}{3} \pi \left(\frac{A}{2}\right)^3 \]
Simplifying this:
\[ V_S = \frac{4}{3} \pi \left(\frac{A^3}{8}\right) = \frac{4 \pi A^3}{24} = \frac{\pi A^3}{6} \]

6. **Compare the Volumes of the Cone and Sphere:**
- Volume of the cone \( V_C = \frac{\pi A^3}{12} \)
- Volume of the sphere \( V_S = \frac{\pi A^3}{6} \)

Now, we can see that:
\[ V_S = 2 \times V_C \]
This means the volume of the sphere is greater than the volume of the cone.

### Conclusion:
Since \( V_S > V_C \), we can conclude that the volume of the sphere is greater than the volume of the cone.