The ratio of the volume of a cube to that of a sphere which will fit inside the cube is

Options

A. `4:pi`
B. `4:3pi`
C. `6:pi`
D. `2:pi`

Answer

To find the ratio of the volume of a cube to that of a sphere that fits inside the cube, we can follow these steps:

### Step 1: Define the side of the cube
Let the side length of the cube be \( A \).

### Step 2: Determine the diameter and radius of the sphere
Since the sphere fits perfectly inside the cube, the diameter of the sphere is equal to the side length of the cube. Therefore, the diameter of the sphere is \( A \). The radius \( r \) of the sphere is half of the diameter:
\[ r = \frac{A}{2} \]

### Step 3: Calculate the volume of the cube
The volume \( V_{\text{cube}} \) of the cube is given by the formula:
\[ V_{\text{cube}} = A^3 \]

### Step 4: Calculate the volume of the sphere
The volume \( V_{\text{sphere}} \) of the sphere is given by the formula:
\[ V_{\text{sphere}} = \frac{4}{3} \pi r^3 \]
Substituting \( r = \frac{A}{2} \) into the volume formula:
\[ V_{\text{sphere}} = \frac{4}{3} \pi \left(\frac{A}{2}\right)^3 = \frac{4}{3} \pi \left(\frac{A^3}{8}\right) = \frac{4}{24} \pi A^3 = \frac{\pi A^3}{6} \]

### Step 5: Find the ratio of the volumes
Now, we can find the ratio of the volume of the cube to the volume of the sphere:
\[ \text{Ratio} = \frac{V_{\text{cube}}}{V_{\text{sphere}}} = \frac{A^3}{\frac{\pi A^3}{6}} = \frac{A^3 \cdot 6}{\pi A^3} = \frac{6}{\pi} \]

### Conclusion
The ratio of the volume of the cube to that of the sphere that fits inside it is:
\[ \frac{6}{\pi} \]

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