Rounds around cylinders with different radii (fixed rope length): A rope can make 70 complete rounds around a cylinder whose base radius is 14 cm. Using the same rope (same length), how many complete rounds will it make around another cylinder whose base radius is 20 cm?

Introduction / Context:
Wrapping a rope around a cylinder relates the rope length to the cylinder’s circumference. For horizontal wraps without gaps or overlaps, one round equals one full circumference. With fixed rope length, the number of rounds is inversely proportional to the circumference and hence inversely proportional to the radius (since C = 2 * pi * r).

Given Data / Assumptions:

• Number of rounds on cylinder 1 (r1 = 14 cm) = 70
• Same rope length used on cylinder 2 (r2 = 20 cm)
• Neglect rope thickness and any gaps; wraps are tight and contiguous.

Concept / Approach:
Let L be the rope length. Then L = n1 * 2 * pi * r1. For the second cylinder, n2 = L / (2 * pi * r2). Substituting L from the first equation shows n2 = n1 * r1 / r2, revealing the inverse proportionality to radius.

Step-by-Step Solution:
L = 70 * 2 * pi * 14n2 = L / (2 * pi * 20) = (70 * 2 * pi * 14) / (40 * pi)n2 = 70 * (14 / 20) = 70 * 0.7 = 49 rounds

Verification / Alternative check:
Proportion shortcut: n ∝ 1 / r → n2 = n1 * r1 / r2 = 70 * 14 / 20 = 49. This matches the full circumference-based derivation.

Why Other Options Are Wrong:
56 and 54 assume smaller radii or longer rope; 77 would require a larger number of wraps than with the smaller radius, which contradicts inverse proportionality; 42 undercounts relative to the precise ratio 14/20.

Common Pitfalls:
Using diameter instead of radius (which would double-count) or forgetting that circumference is directly proportional to radius. Keep pi factors consistent so they cancel correctly.

Final Answer:
49 rounds