A cube is inscribed in a sphere. Inside the cube, a right circular cylinder is placed so that it touches all the vertical faces. What is the ratio of the volume of the cube to the volume of the cylinder?

Introduction / Context:
In this three dimensional mensuration problem, a cube contains a right circular cylinder that touches all of the cube's vertical faces. We are asked to find the ratio of the cube's volume to that of the cylinder. Although the cube is also inscribed in a sphere, that additional information does not change the ratio between cube and cylinder volumes, because both solids are defined entirely by the cube's side length. The solution requires setting up the cylinder's radius and height in terms of the cube side and then simplifying the resulting volume ratio.

Given Data / Assumptions:
- Let the cube have side length a. - The cylinder is inside the cube and touches all four vertical faces. - Therefore, the cylinder's circular base fits exactly into the square face of side a. - Cylinder height is equal to the cube's height, which is a. - We use π and may approximate π as 22 / 7 when expressing the final ratio in whole numbers.

Concept / Approach:
If the cylinder touches all vertical faces of the cube, its diameter equals the side length of the cube. Hence the cylinder radius is r = a / 2 and height h = a. The cube volume is V_cube = a^3, and the cylinder volume is V_cyl = π * r^2 * h. Substituting r and h in terms of a gives V_cyl = (π / 4) * a^3. The ratio V_cube : V_cyl can thus be written symbolically as 1 : (π / 4) or equivalently 4 : π. To match the answer choices given, which are rational ratios of integers, we approximate π by 22 / 7 and then simplify 4 : π to a ratio of whole numbers.

Step-by-Step Solution:
Step 1: Let the cube side length be a. Then cube volume is V_cube = a^3. Step 2: Since the cylinder touches all vertical faces, its diameter equals a, so radius r = a / 2. Step 3: The cylinder height equals the cube height, so h = a. Step 4: Cylinder volume is V_cyl = π * r^2 * h. Step 5: Substitute r and h to get V_cyl = π * (a / 2)^2 * a = π * (a^2 / 4) * a. Step 6: Simplify to obtain V_cyl = (π / 4) * a^3. Step 7: Form the volume ratio V_cube : V_cyl = a^3 : (π / 4) * a^3. Step 8: Cancel a^3 from numerator and denominator to obtain 1 : (π / 4). Step 9: Rewrite as 4 : π by multiplying both parts of the ratio by 4. Step 10: Use π ≈ 22 / 7, so 4 : π ≈ 4 : (22 / 7). Step 11: Multiply both sides of 4 : (22 / 7) by 7 to obtain 28 : 22. Step 12: Simplify 28 : 22 by dividing both numbers by 2 to get 14 : 11.

Verification / Alternative check:
To confirm, take a simple value such as a = 2 units. Then the cube volume is 2^3 = 8 units³. The cylinder radius is 1 unit and height is 2 units, so V_cyl = π * 1^2 * 2 = 2π units³. The exact ratio V_cube : V_cyl is 8 : 2π = 4 : π, as derived earlier. Approximating π as 22 / 7, the ratio becomes approximately 4 : 3.142857..., and converting 4 : (22 / 7) into integers gives 14 : 11. This numerical example is fully consistent with the algebraic derivation.

Why Other Options Are Wrong:
Option 4 : 3 approximates π as exactly 3, which is too rough and does not match the 22 / 7 based simplification. Option 21 : 16 does not simplify to 4 : π under any standard approximation of π. Option 45 : 32 is another arbitrary rational ratio with no link to the factor π / 4 that appears in the cylinder volume.

Common Pitfalls:
Common errors include misinterpreting touching all the vertical faces and using the face diagonal as the cylinder diameter instead of the side length. Another mistake is to forget that the cylinder height is exactly the cube side length. Students might also try to cancel π prematurely or incorrectly handle the ratio algebra, leading to wrong rational approximations. Carefully writing out V_cube and V_cyl in terms of a and simplifying step by step before approximating π helps avoid these issues.

Final Answer:
The ratio of the volume of the cube to the volume of the cylinder is 14 : 11 when expressed using π ≈ 22 / 7.