Thus, the greatest common factor of 68 and 102 is \(\boxed{34}\).**Question:** A linguist studying the evolution of language families models the spread of a new dialect as a circular wave expanding uniformly from a central point. If the wavefront forms a circle with radius increasing by 3 cm per second, what is the area of the region covered by the dialect after 4 seconds? Express your answer in terms of $\pi$.

["Unlocking Geometry and Evolution: A Linguistic Model of Dialect Expansion and Mathematical Precision", "Language evolves like a wave—diffusing outward from a cultural epicenter, adapting, merging, and splitting across communities. In a fascinating interdisciplinary model, a linguist uses geometric principles to simulate the spatial spread of a new dialect. Consider how the expansion of linguistic influence can be mirrored mathematically through the propagation of a circular wave, where distance from a source increases uniformly over time.", "In one such model, the dialect spreads radially from a central point, with the radius growing at a constant rate. If the radius increases by 3 cm per second, after 4 seconds, the wavefront forms a perfect circle. To determine the linguistic reach at that moment, we first compute the radius:", "Radius = rate × time = (3,\ ext{cm/sec} \ imes 4,\ ext{sec} = 12,\ ext{cm})", "Now, with the radius known, we calculate the area of the circular region covered—the spatial domain in which the dialect takes hold. The formula for the area (A) of a circle is:", "[\nA = \pi r^2\n]", "Substituting (r = 12,\ ext{cm}):", "[\nA = \pi (12)^2 = \pi \ imes 144 = 144\pi,\ ext{cm}^2\n]", "Thus, the area of the region influenced by the dialect after 4 seconds is (\boxed{144\pi}) cm².", "This model illustrates how abstract mathematical concepts like geometric area support deep inquiries into cultural diffusion—where language spreads, grows, and transforms, just as circles expand in plane geometry. By connecting linguistic evolution to measurable spatial patterns, we gain clearer insight into the dynamics of human communication across time and space."]