The volumes of a sphere and a right circular cylinder (with the same radius) are equal. Find the ratio of the diameter of the sphere to the height of the cylinder.

Introduction / Context:
Equate volumes of sphere and cylinder with common radius r. Express the cylinder height in terms of r, then form the ratio of sphere diameter to that height.

Given Data / Assumptions:

• V_sphere = (4/3)πr^3.
• V_cyl = πr^2h.
• Equal volumes: V_sphere = V_cyl.

Concept / Approach:
Solve (4/3)πr^3 = πr^2h for h, then compute (diameter):(height).

Step-by-Step Solution:

Verification / Alternative check:
Cancel π and r^2 safely since r > 0; ratio reduces cleanly.

Why Other Options Are Wrong:
They do not reflect h = 4r/3 and D = 2r.

Common Pitfalls:
Using circumference or surface areas instead of volumes; forgetting diameter is 2r.

Final Answer:
3 : 2