Ratio of volumes of two cylinders: Two cylinders have radii in the ratio 2 : 3 and heights in the ratio 5 : 3. Find the ratio of their volumes (first : second).

Introduction / Context:The volume of a cylinder scales as r^2h. For similar or related solids, dimensional ratios translate into volume ratios by squaring the radius ratio and multiplying by the height ratio. This checks proportional reasoning.

Given Data / Assumptions:

• r1 : r2 = 2 : 3
• h1 : h2 = 5 : 3
• V ∝ r^2h

Concept / Approach:Compute (r1^2 : r2^2) = (2^2 : 3^2) = (4 : 9) and multiply by (h1 : h2) = (5 : 3). The resulting ratio is (4*5 : 9*3) = (20 : 27).

Step-by-Step Solution:r^2 ratio = 4 : 9Multiply by h ratio → 4*5 : 9*3 = 20 : 27

Verification / Alternative check:Pick convenient numbers: let r1 = 2, r2 = 3, h1 = 5, h2 = 3. Then V1 = π * 2^2 * 5 = 20π; V2 = π * 3^2 * 3 = 27π → ratio 20 : 27.

Why Other Options Are Wrong:4:9 and 9:4 ignore height; 27:20 reverses order; 10:9 is unrelated to r^2h scaling.

Common Pitfalls:Forgetting to square the radius ratio; mixing up which cylinder is “first” in the final ratio.

Final Answer:20 : 27