Cone height from volume equality with a sphere: A sphere of radius r has the same volume as a cone whose base radius is also r. Find the height of the cone.

Introduction / Context:Equating volumes tests recognition of standard formulas and simple algebra. Equal base radii remove one variable immediately.

Given Data / Assumptions:

• V_sphere = (4/3)πr^3
• V_cone = (1/3)πr^2h
• V_sphere = V_cone

Concept / Approach:Cancel common factors and solve for h in terms of r.

Step-by-Step Solution:(4/3)πr^3 = (1/3)πr^2hMultiply both sides by 3/(πr^2): 4r = hTherefore, h = 4r

Verification / Alternative check:Substitute h = 4r back: V_cone = (1/3)πr^2 * 4r = (4/3)πr^3 = V_sphere, confirming equality.

Why Other Options Are Wrong:2r, r/3, and (2/3)r understate the needed height; 3r still falls short; only 4r equates the volumes.

Common Pitfalls:Forgetting the 1/3 factor in cone volume; mixing radius of sphere with diameter in cone base.

Final Answer:4r