Shortest Troll Bridge Span
The coastline
as shown in this map of a coastal inlet entrance is described by the following equation
where equals feet.
The port city, in brown, needs to build a bridge straight across the inlet entrance. What's the shortest possible span, to the nearest integer number of feet?
The answer is 2676.
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1 solution
2 7 1 4 is the obvious but incorrect answer, going from tip to tip almost vertically. The shorter way goes from the tip to the shore on the left side, with a span of 2 6 7 6 . The graphic below shows where the spans are, including one more that's [sort of] a local maximum, while the other two are local minimums
A necessary condition for a straight span to be a local minimum is that it be perpendicular to the shoreline at both ends.
If the curve can be described as a single valued function
y = f ( x )
then the x coordinates x 1 and x 2 of the ends of the bridge spans can be found as solutions of this set of simultaneous equations
− f ′ ( x 1 ) 1 = − f ′ ( x 2 ) 1
x 2 − x 1 f ( x 2 ) − f ( x 1 ) = − f ′ ( x 1 ) 1
which for the given equation has 3 real pairs of solutions,
− 1 . 2 9 0 9 6 9 3 0 8 2 , 1 . 2 0 9 8 9 5 5 4 0 6
− 0 . 2 1 1 2 3 5 9 5 5 0 , 1 . 2 6 7 3 6 5 9 4 2 3
0 . 6 1 2 3 6 7 1 9 5 2 , 1 . 2 8 0 9 1 5 4 8 1 8
of which the first has the shortest length.