The height and base radius of a right circular cylinder are 9 cm and 6 cm…

The volume of a right circular cylinder is V = π × r2 × h, where r is the base radius and h is the height. Because the radius appears squared in this formula, a change to r affects the volume by the square of that change, while a change to h affects it only in direct proportion — so when both r and h change together, the new volume must be found by substituting the new values, not by intuition about which change "seems bigger".

1. Original dimensions: radius r1 = 6 cm, height h1 = 9 cm. Original volume V1 = π × 62 × 9 = π × 36 × 9 = 324π cm3.

2. New radius: half of the original radius = 6 ÷ 2 = 3 cm.

3. New height: original height increased by 3 cm = 9 + 3 = 12 cm.

4. New volume: V2 = π × 32 × 12 = π × 9 × 12 = 108π cm3.

5. Ratio of new volume to original volume: V2 ÷ V1 = 108π ÷ 324π = 108 ÷ 324 = 1/3.

Cross-check with scale factors directly: the radius scale factor is 3 ÷ 6 = 1/2, so the base area (which depends on r2) scales by (1/2)2 = 1/4; the height scale factor is 12 ÷ 9 = 4/3. Multiplying the two scale factors gives the overall volume scale factor: 1/4 × 4/3 = 1/3 — the same result as the direct substitution above.

Therefore, the new volume works out to exactly one-third of the original volume.