The radii of two cones are in the ratio 2 : 1 and their volumes are equal. The…

Concept: For a cone of radius r and height h, volume V = (1/3)×π×r2×h. When two cones have equal volumes, the common factor (1/3)π cancels from both sides, leaving r2h as an invariant quantity — so, for a fixed volume, the height of a cone varies inversely as the square of its radius.

Application:

1. Let the radii of the two cones be r1 and r2, with r1 : r2 = 2 : 1, so r1 = 2k and r2 = k for some common unit k.

2. The volumes are equal, so (1/3)πr12h1 = (1/3)πr22h2. Cancelling (1/3)π from both sides gives r12h1 = r22h2.

3. Substituting r1 = 2k and r2 = k: (2k)2h1 = (k)2h2, i.e. 4k2h1 = k2h2.

4. Cancelling k2 from both sides: 4h1 = h2, so h1 / h2 = 1 / 4.

5. Therefore h1 : h2 = 1 : 4.

Cross-check: Take r1 = 2, r2 = 1, h1 = 1, h2 = 4 (any common unit works). Volume1 = (1/3)π(2)2(1) = (4/3)π; Volume2 = (1/3)π(1)2(4) = (4/3)π. The two volumes are equal, confirming the height ratio.

Result: The ratio of the heights is 1 : 4.