Question

Determine the ratio of the volume of a cube to that of the sphere which will exactly fit inside the cube.

Answer 1

Step 1: Understanding the problem:
We are given a cube and a sphere that fits exactly inside the cube. Our task is to determine the ratio of the volume of the cube to the volume of the sphere.

Step 2: Formula for the volume of the cube:
The volume of a cube is given by:
\[ V_{\text{cube}} = a^3 \] where \( a \) is the side length of the cube.

Step 3: Formula for the volume of the sphere:
The volume of a sphere is given by:
\[ V_{\text{sphere}} = \frac{4}{3} \pi r^3 \] where \( r \) is the radius of the sphere.

Step 4: Relationship between the cube and the sphere:
Since the sphere fits exactly inside the cube, the diameter of the sphere is equal to the side length of the cube. Therefore, we have:
\[ 2r = a \] or equivalently, \[ r = \frac{a}{2} \]

Step 5: Volume of the sphere in terms of \( a \):
Substituting \( r = \frac{a}{2} \) into the volume formula for the sphere:
\[ V_{\text{sphere}} = \frac{4}{3} \pi \left( \frac{a}{2} \right)^3 = \frac{4}{3} \pi \frac{a^3}{8} = \frac{\pi a^3}{6} \]

Step 6: Ratio of the volumes:
The ratio of the volume of the cube to the volume of the sphere is:
\[ \text{Ratio} = \frac{V_{\text{cube}}}{V_{\text{sphere}}} = \frac{a^3}{\frac{\pi a^3}{6}} = \frac{6}{\pi} \]

Conclusion:
The ratio of the volume of the cube to the volume of the sphere is \( \frac{6}{\pi} \).