Two cylindrical buckets have their diameters in the ratio 3 : 1 and their heights in the ratio 1 : 3. What is the ratio of their volumes?

Introduction / Context:
This question examines proportional reasoning and the volume formula for cylinders. Instead of giving actual dimensions, it provides ratios of diameters and heights of two cylindrical buckets. Candidates must know that the volume of a cylinder depends on the square of the radius and the height, and then apply ratio rules correctly. Such questions are frequent in aptitude tests dealing with similar solids.

Given Data / Assumptions:
• Ratio of diameters of the two buckets = 3 : 1.• Therefore, ratio of radii is also 3 : 1.• Ratio of heights of the two buckets = 1 : 3.• We are asked to find the ratio of their volumes.

Concept / Approach:
The volume of a cylinder is V = π * r^2 * h, where r is radius and h is height. When comparing two cylinders, we can use ratios rather than actual numbers. If r1 : r2 = 3 : 1, then r1^2 : r2^2 = 9 : 1. If h1 : h2 = 1 : 3, we combine these to get the volume ratio: V1 : V2 = (r1^2 * h1) : (r2^2 * h2). Constants like π cancel out in the ratio, making calculations simpler.

Step-by-Step Solution:
Step 1: Let the radii be r1 and r2 with r1 : r2 = 3 : 1.Then r1^2 : r2^2 = 9 : 1.Step 2: Let the heights be h1 and h2 with h1 : h2 = 1 : 3.Step 3: Write the volume ratio.V1 : V2 = (π * r1^2 * h1) : (π * r2^2 * h2).Step 4: Cancel π from the ratio.V1 : V2 = (r1^2 * h1) : (r2^2 * h2).Step 5: Substitute the ratios.V1 : V2 = 9 * 1 : 1 * 3 = 9 : 3.Step 6: Simplify the ratio.9 : 3 = 3 : 1.

Verification / Alternative check:
We can assign actual values that respect the ratio. For example, let bucket 1 have radius 3 units and height 1 unit, and bucket 2 have radius 1 unit and height 3 units. Then V1 = π * 3^2 * 1 = 9π and V2 = π * 1^2 * 3 = 3π. The ratio 9π : 3π simplifies to 3 : 1, confirming the result.

Why Other Options Are Wrong:
Option A: 2:1 does not reflect the combined effect of squaring the radius and the height ratio.Option C: 4:1 would require different diameter or height ratios.Option D: 5:1 is not supported by any correct combination of the given ratios.

Common Pitfalls:
Some learners forget to square the radius and treat volume as proportional to r * h instead of r^2 * h. Others might mistakenly multiply the diameter ratio by the height ratio directly, getting 3:1 * 1:3 = 1:1, which is wrong. Remembering that volume for a cylinder involves the square of the radius is crucial to avoid these errors.

Final Answer:
The ratio of the volumes of the two buckets is 3:1.