The radii of two cylinders are in the ratio 3 : 2 and their heights are in the ratio 3 : 7. What is the ratio of their volumes?

Introduction / Context:
This question checks the understanding of how the volume of a cylinder depends on its radius and height. We are given the ratios of the radii and heights of two cylinders and asked to find the ratio of their volumes. It illustrates how to combine given ratios and apply them to formula based relationships, an important skill in quantitative reasoning.

Given Data / Assumptions:
• Let the radii of the two cylinders be r₁ and r₂ with r₁ : r₂ = 3 : 2.
• Let their heights be h₁ and h₂ with h₁ : h₂ = 3 : 7.
• Volumes of the cylinders are V₁ and V₂ respectively.
• We need the ratio V₁ : V₂.

Concept / Approach:
The volume of a right circular cylinder is V = πr²h. For two cylinders, we form the ratio V₁ / V₂ = (πr₁²h₁) / (πr₂²h₂). The π cancels, leaving a ratio involving r₁² : r₂² and h₁ : h₂. We use the given simple ratios to express r₁ and r₂ and h₁ and h₂ in proportional terms, and then compute the resulting ratio of volumes. Finally, we simplify the fraction to lowest terms.

Step-by-Step Solution:
Step 1: Use the ratio r₁ : r₂ = 3 : 2. We can take r₁ = 3k and r₂ = 2k for some positive k. Step 2: Use the ratio h₁ : h₂ = 3 : 7. We can take h₁ = 3m and h₂ = 7m for some positive m. Step 3: Volume of cylinder 1 is V₁ = π r₁² h₁ = π (3k)² (3m) = π * 9k² * 3m = 27πk²m. Step 4: Volume of cylinder 2 is V₂ = π r₂² h₂ = π (2k)² (7m) = π * 4k² * 7m = 28πk²m. Step 5: Form the ratio V₁ : V₂ = 27πk²m : 28πk²m. Step 6: Cancel π, k², and m from numerator and denominator to get 27 : 28.

Verification / Alternative check:
We can also substitute specific values consistent with the ratios, such as r₁ = 3, r₂ = 2, h₁ = 3, and h₂ = 7. Then V₁ = π * 3² * 3 = π * 9 * 3 = 27π and V₂ = π * 2² * 7 = π * 4 * 7 = 28π. The ratio 27π : 28π simplifies directly to 27 : 28. This confirms the earlier symbolic calculation and shows that the exact proportional constants k and m do not affect the ratio.

Why Other Options Are Wrong:
4 : 7 and 7 : 4 come from combining radii and heights incorrectly, for example by ignoring the square on the radius or by adding exponents wrongly.
28 : 27 is the reverse of the correct ratio, implying the second cylinder is slightly smaller, which is not supported by the calculated volumes.
9 : 7 reflects only the square of the radius ratio (3² : 2²) but ignores the height ratio, so it is incomplete and incorrect.

Common Pitfalls:
A typical error is to forget that the radius is squared in the volume formula, leading to using r instead of r² in the ratio. Another pitfall is mixing up the order of the ratio or failing to incorporate the height ratio. Some learners attempt to add or multiply ratios incorrectly without grounding them in the formula. The correct approach is to substitute proportional values and compute the resulting volumes before forming the ratio.

Final Answer:
The ratio of the volumes of the two cylinders is 27 : 28.