Solution: Since the cones are similar, the ratio of their linear dimensions is $ \frac{15}{60} = \frac{1}{4} $. The volume ratio of similar solids is the cube of the linear ratio:

["Solution: Volume Ratio of Similar Cones Based on Linear Dimension Ratios", "Understanding the relationship between linear dimensions and volume in similar solids is essential in geometry and applied mathematics. When two cones are considered similar—meaning their shapes are identical but sizes differ—the ratio of their corresponding linear dimensions determines how their volumes compare.", "---", "### The Foundation: Similarity and Linear Ratios", "In geometry, if two solids are similar, all their corresponding linear measurements (such as radius, height, or slant height) are proportional. The problem states that the ratio of the linear dimensions of the cones is:", "[\n\frac{15}{60} = \frac{1}{4}\n]", "This indicates the smaller cone is reduced to one-fourth the size of the larger cone in every linear respect.", "---", "### Volume Ratio: The Cube of the Linear Ratio", "While linear dimensions scale uniformly in similar solids, volume is a three-dimensional measure and depends on the cube of the linear scale factor. This is a key principle in geometric similarity:", "[\n\ ext{Volume Ratio} = \left( \frac{\ ext{Linear Ratio}}{} \right)^3\n]", "Substituting the given ratio:", "[\n\ ext{Volume Ratio} = \left( \frac{1}{4} \right)^3 = \frac{1}{64}\n]", "---", "### Conclusion", "Therefore, the volume of the smaller cone is one sixty-fourth the volume of the larger cone. This relationship holds true for any pair of similar cones — not just the ones referenced here — as long as their linear dimensions maintain the given ratio. Recognizing this cube relationship allows efficient computation of volume without full measurement, demonstrating the power of geometric similarity in problem-solving.", "---", "Key Takeaway:\nFor similar three-dimensional solids, the ratio of their volumes equals the cube of the ratio of their corresponding linear dimensions. In this case, the volume ratio is $ \frac{1}{64} $.", "---", "Summary:\n- Linear ratio: $ \frac{15}{60} = \frac{1}{4} $\n- Volume ratio = $ \left( \frac{1}{4} \right)^3 = \frac{1}{64} $\n- More complex cone problems often rely on this cube relationship to simplify volume comparisons."]