A cone and a cylinder have equal heights. The ratio of the radii of their bases (cone : cylinder) is 2 : 1. What is the ratio of the volume of the cone to the volume of the cylinder?

Introduction / Context:
This problem tests your understanding of the volumes of a right circular cone and a right circular cylinder when their heights are equal. Both solids are fundamental three dimensional shapes in mensuration. Many aptitude questions compare their volumes when certain dimensions are related through simple ratios. Knowing the volume formulas and being able to substitute ratios instead of actual numbers allows you to solve such problems quickly and without complex calculations.

Given Data / Assumptions:
• There is a right circular cone and a right circular cylinder.
• The heights of the cone and the cylinder are equal; let this common height be h units.
• The ratio of their radii (cone : cylinder) is 2 : 1. Let the radius of the cone be 2r and the radius of the cylinder be r.
• We are asked to find the ratio of the volume of the cone to the volume of the cylinder.

Concept / Approach:
Recall the standard volume formulas:
• Volume of a cylinder: V_cylinder = π * r^2 * h.
• Volume of a cone: V_cone = (1 / 3) * π * R^2 * h, where R is the radius of the cone and h is the height. Since both solids have the same height and radii in ratio 2 : 1, we can express the radii using a common variable r and then compute the ratio of volumes symbolically. This approach keeps the calculations simple and independent of specific numerical heights or radii.

Step-by-Step Solution:
Step 1: Let the radius of the cylinder be r and its height be h. Step 2: Given that the radii are in the ratio 2 : 1 (cone : cylinder), the radius of the cone is 2r. Step 3: Write the volume of the cone: V_cone = (1 / 3) * π * (2r)^2 * h = (1 / 3) * π * 4r^2 * h = (4 / 3) * π * r^2 * h. Step 4: Write the volume of the cylinder: V_cylinder = π * r^2 * h. Step 5: Form the ratio V_cone : V_cylinder = (4 / 3) * π * r^2 * h : π * r^2 * h. Step 6: Cancel the common factors π, r^2 and h to get V_cone : V_cylinder = 4 / 3 : 1 = 4 : 3.

Verification / Alternative check:
You can choose simple numerical values to verify. For example, let r = 1 unit and h = 3 units. Then the cylinder's volume is π * 1^2 * 3 = 3π. The cone's radius is 2 units, so its volume is (1 / 3) * π * 2^2 * 3 = (1 / 3) * π * 4 * 3 = 4π. The ratio V_cone : V_cylinder is 4π : 3π, which simplifies to 4 : 3, matching our algebraic result. This confirms that the ratio does not depend on the specific numerical values of r and h as long as the given conditions are met.

Why Other Options Are Wrong:
A ratio of 3 : 4 would imply that the cone has smaller volume than the cylinder, which is not true when the cone's radius is larger and only the height is scaled in the same way. Ratios of 2 : 1 or 1 : 2 ignore the factor 1 / 3 present in the cone volume formula and the squaring of the radius, leading to incorrect comparisons. Only 4 : 3 correctly accounts for both the squared radius ratio and the 1 / 3 factor in the cone volume formula.

Common Pitfalls:
A common mistake is to forget the 1 / 3 factor in the cone volume formula and to write V_cone = π * R^2 * h instead of (1 / 3) * π * R^2 * h. Another error is to treat the radii linearly without squaring them, which would give a wrong ratio. Always remember that in volume formulas for solids of revolution, the radius usually appears squared (as in π * r^2), so ratio comparisons must square the scale factor for radii. Checking the formula carefully before substituting ratios helps avoid these errors.

Final Answer:
The ratio of the volume of the cone to the volume of the cylinder is 4:3.