In a swim-and-run biathlon, an athlete must get to a point on the other side of a 50 meter wide river, 100 meters downstream from her starting point. ann can swim 2 m/sec and run 5 m/sec. what path should ann take in order to minimize her total time?

running distance =   78,18 m

swimmingdistance  =  92m

Step-by-step explanation:

Ann has to run a distance 100 - x    and swim  √ (50)² + x²

at speed of 5 m/sec   and 2 m/sec

As distance  = v*t       t  = d/v

Then running she will spend time doing d = ( 100-x)/5    

and   √[(50)² + x² ] / 2   swimming

Therefore total time of biathlon

t(x)  =  ( 100 - x )/5    +  √[(50)² + x² ] / 2

Taking derivatives both sides of the equation we get

t´(x)  =  - 1/5  + [1/2 ( 2x)*2] / 4√[(50)² + x²]

t´(x)  =  - 1/5  + 2x / 4√[(50)² + x²]      t´(x)  =  - 1/5  + x/2√(50)² + x²

t´(x)  = 0           - 1/5  + x/2√(50)² + x²  = 0

  - 2√[(50)²+  x²]   +  5x     =  0

  - 2√(50)²+  x² ) =  -5x

       √(50)²+  x²   = 5/2 *x

squared

        (50)²  +  x²   = 25/4 x²

             2500 - 21/4 x²  =  0

                           x²  =  2500*4/21

 x  =  21,8 m

Therefore she has to run  100 - 21,82  = 78,18 m

And swim    √(50)² + (78,18)²   =  92m

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