Final answer:
The detail showing how the historical setting influences preparations is the family dressing in their finest attire to witness the engineering marvel of the Great East River Bridge, signifying a moment of pride and progress in the historical era.
Step-by-step explanation:
The detail that best shows how the historical setting influences the narrator’s preparations for the day is the family putting on their “finest dresses” and “shoes Mother had polished twice” to walk on the newly built Great East River Bridge, which we now know as the Brooklyn Bridge. This indicates a significant event in the period, meriting special attire and reflecting the pride and excitement of witnessing an engineering marvel of their time. The bridge symbolizes progress and the overcoming of various challenges (“deals gone bad, workers with illnesses, and defiance of something called physics”), suggesting a historical moment of achievement and celebration for the people during that era.
References to the bridge in various texts, including the description of the bridge as a “grand archway” and seeing the bridge from a distance as in the passage about the Hudson River's scenery, highlight the iconic status and influence of such structures in the American consciousness. Bridges are not only functional but also emblematic of the era’s engineering feats and are woven into the narrative of the nation's development and identity.