A cube is inscribed in a sphere. Inside the cube, a right circular cylinder is placed so that it touches all the vertical faces. Inside this cylinder, a right circular cone is placed with the same height as the cylinder and with diameter equal to that of the cylinder. What is the ratio of the volume of the sphere to the volume of the cone?

Introduction / Context:
This is a three dimensional geometry question involving several nested solids: a sphere, a cube, a cylinder and a cone. Each solid is inscribed in the previous one in a specific way. We are asked to find the ratio of the volume of the largest solid (the sphere) to the smallest solid (the cone). Although the configuration may seem complex, the key observation is that all dimensions can be expressed in terms of a single parameter, the side length of the cube. Once the relationships between the radii and heights are established, volume formulas lead to a simple ratio that no longer depends on the actual size of the cube.

Given Data / Assumptions:
- A cube of side length a is inscribed in a sphere, so all its vertices lie on the sphere. - A right circular cylinder is inscribed in the cube touching all four vertical faces. - A right circular cone is inscribed in the cylinder with the same height as the cylinder. - The cone has the same diameter as the cylinder, so their radii are equal. - We assume all solids are solid, not hollow, and share the same central vertical axis.

Concept / Approach:
First, express the radius of the sphere in terms of the cube side length a. For a cube inscribed in a sphere, the sphere radius equals half the space diagonal of the cube. The space diagonal of a cube with side a is a * √3, so the sphere radius is (a * √3) / 2. Next, the cylinder inscribed in the cube touches all vertical faces, so its circular base fits exactly into the square face of side a, giving cylinder radius r_cyl = a / 2 and cylinder height h_cyl = a. The cone inside the cylinder has the same height and radius as the cylinder, so r_cone = a / 2 and h_cone = a. We then use the volume formulas for sphere and cone to form the required ratio.

Step-by-Step Solution:
Step 1: Let the cube side length be a. Step 2: The space diagonal of the cube is a * √3, so the radius of the sphere is R = (a * √3) / 2. Step 3: The cylinder is inscribed in the cube with its base touching all four vertical faces, so its radius is r = a / 2 and its height is h = a. Step 4: The cone shares the same radius and height as the cylinder, so r_cone = a / 2 and h_cone = a. Step 5: Volume of the sphere is V_sphere = (4 / 3) * π * R^3. Step 6: Substitute R = (a * √3) / 2 to get R^3 = (a^3 * 3√3) / 8. Step 7: So V_sphere = (4 / 3) * π * (a^3 * 3√3 / 8) = (4 * 3√3 / 24) * π * a^3 = (√3 / 2) * π * a^3. Step 8: Volume of the cone is V_cone = (1 / 3) * π * r_cone^2 * h_cone. Step 9: Substitute r_cone = a / 2 and h_cone = a to get V_cone = (1 / 3) * π * (a^2 / 4) * a = (1 / 12) * π * a^3. Step 10: Form the ratio V_sphere : V_cone = ((√3 / 2) * π * a^3) : ((1 / 12) * π * a^3). Step 11: Cancel π and a^3 from numerator and denominator to obtain (√3 / 2) : (1 / 12). Step 12: Divide (√3 / 2) by (1 / 12) to get (√3 / 2) * 12 = 6√3. Step 13: Therefore the ratio of the volume of the sphere to that of the cone is 6√3 : 1.

Verification / Alternative check:
To check, we can assign a numerical value, say a = 2 units. Then R = √3, r_cone = 1 and h_cone = 2. The sphere volume is V_sphere = (4 / 3) * π * (√3)^3 = (4 / 3) * π * 3√3 = 4π√3. The cone volume is V_cone = (1 / 3) * π * 1^2 * 2 = (2 / 3)π. The ratio V_sphere / V_cone equals (4π√3) / ((2 / 3)π) = 4√3 * (3 / 2) = 6√3, consistent with the symbolic computation. This confirms that the dimension choice does not affect the final ratio.

Why Other Options Are Wrong:
Option 7 : 2 is a rational ratio and ignores the presence of √3 that arises from the cube diagonal. Option 3√3 : 1 is exactly half of the correct ratio and would result from mistakenly using diameter instead of radius in the sphere volume. Option 5√3 : 1 has no direct relation to the computed volumes and does not match any consistent geometric simplification.

Common Pitfalls:
Students often misinterpret the phrase touching all the vertical faces and assign the cylinder radius as half the face diagonal instead of half the cube side. Another error is to use the cube side as the sphere radius rather than half the space diagonal. Forgetting to cube the radius correctly in the volume formula or mismanaging the factor of (1 / 3) in the cone volume can also lead to incorrect ratios. Careful identification of each solid's dimensions in terms of a single parameter a is crucial.

Final Answer:
The ratio of the volume of the sphere to the volume of the cone is 6√3 : 1.