A cone and a cylinder are of equal volume. The base radius of the cone is twice the radius of the cylinder…

Analyze the question: A cone and a cylinder have equal volumes. The cone's radius is twice the cylinder's radius. We need the ratio of the cylinder's height to the cone's height.

Concept:

• Volume of cylinder = πr²h
• Volume of cone = (1/3)πR²H

Solution:

Let the cylinder have radius r and height h. The cone then has radius R = 2r and height H.

Equal volumes:

πr²h = (1/3)π(2r)²H

πr²h = (1/3)π(4r²)H

Cancel πr² from both sides:

h = (4/3)H

So h : H = 4 : 3

Check: If H = 3, then h = 4. Cone volume = (1/3)π(4r²)(3) = 4πr². Cylinder volume = πr²(4) = 4πr². They match.

Exam tip: In "equal volume" problems, write both formulas, set them equal, and cancel the common factors (π and r²). Remember that the cone's 1/3 combined with the squared radius ratio (2² = 4) gives 4/3.