The volume of a sphere is proportional to the cube of its radius. If the radii are in the ratio 5:7, then the volumes are in the ratio 5³:7³ = 125:343. Cubing both terms of the radius ratio correctly scales the measure from a linear dimension to a three-dimensional one. Therefore, the ratio of the volumes is 125:343.
Option A:
Option A, 5:7, simply repeats the ratio of the radii and ignores the fact that volume is a three-dimensional measure. Using this ratio would be correct if we were comparing lengths, not volumes, so it is too small a difference for the situation described.
Option B:
Option B, 25:49, is obtained by squaring the radius ratio and is appropriate for comparing surface areas, not volumes. This reflects a two-dimensional rather than a three-dimensional relationship, so 25:49 does not apply here.
Option C:
Option C, 125:196, mixes a cubed numerator with a squared denominator, which has no geometric justification. Such a mismatch does not correspond to any standard dimensional comparison and cannot represent the ratio of volumes.
Option D:
Option D is correct because it cubes both terms of the radius ratio consistently, giving 5³:7³ = 125:343. This matches the known formula V ∝ r³ and correctly captures how volume scales with changes in radius.
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