The radii of two similar cylindrical jugs are in the ratio 3:7. Calculate the ratio of their volumes

Analyze the question: Two similar cylinders have radii in the ratio 3 : 7. We need the ratio of their volumes.

Concept: For similar solids, if the ratio of corresponding lengths (radius, height, etc.) is a : b, then:

• Ratio of areas = a² : b²
• Ratio of volumes = a³ : b³

This works because similar cylinders have all linear dimensions in the same ratio, so the heights are also in the ratio 3 : 7.

Solution:

Volume of a cylinder = πr²h

• Cylinder 1: V₁ = π(3k)²(3k) = 27πk³
• Cylinder 2: V₂ = π(7k)²(7k) = 343πk³

V₁ : V₂ = 27 : 343

Check the options:

• A (1 : 04) and B (3 : 07) are not the cube ratio. B is just the radius ratio.
• C (9 : 49) is the area ratio, which is a common trap.
• D (0.0861...) is not in ratio form. Note that 27/343 ≈ 0.0787, so D does not match either.
• E (27 : 343) is correct.

Tip: Remember "Length → Area → Volume = power 1 → 2 → 3". If the question says similar solids, cube the length ratio for volume. Watch out for option C, because examiners often include the squared ratio as a distractor.