The heights of two right circular cylinders are in the ratio 3 : 5 and their volumes are in the ratio 27 : 80. What is the ratio of their radii?

Introduction / Context:
This problem involves comparing two right circular cylinders using ratios of their heights and volumes. The goal is to deduce the ratio of their radii. Because volume of a cylinder depends on both radius and height, ratios of volume are influenced by ratios of both dimensions. The question tests algebraic manipulation and understanding of how geometric quantities scale.

Given Data / Assumptions:

• Heights of cylinders: h1 : h2 = 3 : 5.
• Volumes: V1 : V2 = 27 : 80.
• Radii of cylinders: r1 and r2, ratio required r1 : r2.
• Both shapes are right circular cylinders.

Concept / Approach:
Volume of a cylinder is given by:
V = pi * r^2 * h Thus:
V1 / V2 = (pi * r1^2 * h1) / (pi * r2^2 * h2) = (r1^2 * h1) / (r2^2 * h2) We substitute the given ratios for V1 : V2 and h1 : h2, and then solve for r1^2 / r2^2. Finally, we take square roots to obtain the ratio r1 : r2.

Step-by-Step Solution:
Given V1 / V2 = 27 / 80. Given h1 / h2 = 3 / 5. Using V proportionality: V1 / V2 = (r1^2 * h1) / (r2^2 * h2). So 27 / 80 = (r1^2 * h1) / (r2^2 * h2). Substitute h1 / h2 = 3 / 5: 27 / 80 = (r1^2 / r2^2) * (3 / 5). Rearrange: r1^2 / r2^2 = (27 / 80) * (5 / 3). Compute: (27 * 5) / (80 * 3) = 135 / 240 = 9 / 16. Thus r1^2 / r2^2 = 9 / 16. Taking square roots: r1 / r2 = 3 / 4.

Verification / Alternative check:
We can choose convenient values based on the ratio. Let h1 = 3k, h2 = 5k, r1 = 3m, r2 = 4m. Then V1 = pi * 3^2 * 3k = 27k * pi and V2 = pi * 4^2 * 5k = 80k * pi. Therefore V1 : V2 = 27k * pi : 80k * pi = 27 : 80, matching the given ratio. This confirms that r1 : r2 = 3 : 4 is consistent with all the information.

Why Other Options Are Wrong:
Options 1 : 3 and 2 : 1 ignore the squared relationship between radius and volume. Ratios like 4 : 7 or 5 : 4 do not produce the given volume ratio when combined with the height ratio 3 : 5. Only 3 : 4 satisfies the volume and height relationships simultaneously.

Common Pitfalls:
Students often miss the square on the radius in the volume formula and treat volume as proportional to r * h instead of r^2 * h. Another common mistake is to invert one of the ratios accidentally when cross-multiplying, leading to an incorrect radius ratio. Writing all steps clearly and keeping track of numerators and denominators helps avoid such errors.

Final Answer:
The ratio of the radii of the two cylinders is 3 : 4.