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One of my favorite problems, before I retired, was the "drop a stone
down a well and listen for the splash." Calculate the well depth
from the time delay. Following the normal sig figs rule eliminates
the answer. Students did not know what to do next!
"Expand" the square root (from the quadratic formula). This shows
precisely what is happening. The "big" terms cancel out - exactly -
leaving the answer in the (quickly converging) series. Sig figs
takes on a whole new meaning in this problem after the expanding
algebra is complete.
And yet I'd still be flabbergasted.