Final answer:
The question involves comparing ratios of base areas, surface areas, and volumes of similar cones, which are determined by the square and cube of the scale factor between the corresponding dimensions of the cones. It also encompasses understanding geometric formulas and the factor-label method for dimensional analysis.
Step-by-step explanation:
The question deals with comparing the ratios of the base areas, surface areas, and volumes of two similar cones. Knowing that similar cones have proportional dimensions, we can determine these ratios. If the scale factor for the radii of the two cones is a:b, then the ratio of their base areas, which are circles, would be a²:b². The surface area involves the base and the lateral surface, so its ratio would also be a²:b². Lastly, the ratio of the volumes of cones, which is determined by the formula 1/3πr²h, would be a³:b³, given that the heights are also proportional with the same scale factor.
In a broader context on dimensional analysis, the question touches upon the comparison of areas and volumes of shapes having the same volume but differing in other dimensions, appropriate use of formulas, and understanding the factor-label method for units conversion. The example provided about the cylinder (volume = πr²h, surface area = 2πr² + 2πrh) serves to illustrate that understanding geometric formulas plays a vital role in figuring out these comparisons.