If the diameters of two right circular cylinders are in the ratio 3 : 2 and their volumes are equal, what is the ratio of the height of the larger-diameter cylinder to the height of the smaller-diameter cylinder?

Introduction / Context:
This question involves right circular cylinders and compares the effect of different diameters on height when the volumes are equal. It tests your understanding of the cylinder volume formula and your ability to manipulate ratios. Such problems are important in aptitude tests because they require both algebraic skill and correct handling of proportional reasoning.

Given Data / Assumptions:

- There are two right circular cylinders, called Cylinder 1 and Cylinder 2.
- Ratio of their diameters is 3 : 2, so d1 : d2 = 3 : 2.
- Volumes of the two cylinders are equal.
- We need the ratio of the height of the larger diameter cylinder to the height of the smaller diameter cylinder, that is h1 : h2.
- π is the usual constant in the volume formula.

Concept / Approach:
The volume of a cylinder is given by V = π * r^2 * h, where r is the radius and h is the height. Since diameters are in ratio 3 : 2, radii are also in ratio 3 : 2. By equating the volumes of the two cylinders and substituting r1 : r2, we can find a relationship between h1 and h2. Simplifying this relationship yields the desired ratio of heights.

Step-by-Step Solution:
Step 1: Let radii be r1 and r2 and heights be h1 and h2 respectively.Step 2: Given diameter ratio 3 : 2, we have r1 : r2 = 3 : 2.Step 3: Volume of Cylinder 1, V1 = π * r1^2 * h1. Volume of Cylinder 2, V2 = π * r2^2 * h2.Step 4: Given volumes are equal, so π * r1^2 * h1 = π * r2^2 * h2.Step 5: Cancel π from both sides to get r1^2 * h1 = r2^2 * h2.Step 6: Using r1 : r2 = 3 : 2, we have r1^2 : r2^2 = 9 : 4.Step 7: Therefore, 9 * h1 = 4 * h2, which gives h1 / h2 = 4 / 9, so h1 : h2 = 4 : 9.

Verification / Alternative check:
Assign simple radii values consistent with the ratio. For instance, let r1 = 3 units and r2 = 2 units. Choose an arbitrary height for one cylinder, say h1 = 4 units. Its volume is V1 = π * 3^2 * 4 = 36π. To match this volume, set V2 = π * 2^2 * h2 = 4π * h2 = 36π, which yields h2 = 9 units. Hence heights are 4 and 9, giving ratio 4 : 9, which confirms the algebraic result.

Why Other Options Are Wrong:
Ratios 2 : 3 and 3 : 2 come from mistakenly inverting the relationship between radius and height or misapplying the square on the radius. Ratios 9 : 4 and 1 : 1 would imply that a larger radius cylinder also has equal or greater height while maintaining equal volume, which contradicts the requirement that a larger base area must be compensated by a smaller height. Only 4 : 9 maintains equal volume with the given diameter ratio.

Common Pitfalls:
Many students forget that the radius appears squared in the cylinder volume formula and directly write h1 : h2 equal to the inverse of the diameter ratio. Others cancel incorrectly or invert the final ratio, giving 9 : 4 instead of 4 : 9. Always remember to square the radius before comparing and carefully check which cylinder is associated with which height in the final ratio.

Final Answer:
The ratio of the height of the larger diameter cylinder to the height of the smaller diameter cylinder is 4 : 9.