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Help with what I think is a differential problem

  1. Nov 3, 2011 #1
    Hi All,

    Did my maths B exam the other day but was absolutely stumped by one question. Spent half an hour thinking about it only to give up.

    Question:

    A man on a kayak (K) is 3 kilometres out to sea from the nearest point, (O) of a straight beach. His destination (D) is 6 kilometres along the beach from O. The fastest he can paddle is 4 km/hr and his maximum walking speed is 5km/h. How far from O should he go ashore to reach his destination in the least possible time?

    Anyways the question was alongside a whole heap of differential problems so I assume you would need to differentiate an equation, find the min SP and that should tell you the distance but I have no idea how to get to the equation or if I was even on the right track.

    This was the only question I couldn't answer and it is driving me nuts.

    Thanks,

    Perjac
     
  2. jcsd
  3. Nov 3, 2011 #2

    HallsofIvy

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    Let P be the point at which the kayak lands on the beach. Let x be the distance from O to P. By the Pythagorean theorem, he must have paddled a distance [itex]\sqrt{x^2+ 9}[/itex] kilometers. At 4 km/hr, that will require [itex](1/4)\sqrt{x^2+9}[/itex] hours. He then has to walk 6- x kilometers. At 5 km/hr, that will require [itex](1/5)(6- x)[/itex] hours. The total time is [itex](1/4)\sqrt{x^2+ 9}+ (1/5)(6- x)[/itex] hours. You want to minimize that. Take the derivative with respect to x and set it equal to 0.
     
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