Unformatted text preview: A to the closest point C on the road. Let v be your speed on the road, and let w , which is less than v , be your speed on the sand. Right now you are at the point D , which is a distance a from C . At what point B should you turn oﬀ the road and head across the sand in order to minimize your travel time to A ? Let x be the distance short of C where you turn oﬀ, i.e., the distance from B to C . We want to minimize the total travel time. Recall that when traveling at constant velocity, time=distance/velocity. You travel the distance DB at speed v , and then the distance BA at speed w . Since DB = ax and, by the Pythagorean theorem, BA = √ x 2 + b 2 , the total time for the trip is f ( x ) = ax v + √ x 2 + b 2 w ....
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 Fall '07
 JonathanRogawski
 Math, Calculus, Critical Point, Pythagorean Theorem, Optimization, ........., π

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