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?

Answer:running distance =   78,18 mswimmingdistance  =  92mStep-by-step explanation:Ann has to run a distance 100 - x    and swim  √ (50)² + x²at speed of 5 m/sec   and 2 m/secAs distance  = v*t       t  = d/vThen running she will spend time doing d = ( 100-x)/5    and   √[(50)² + x² ] / 2   swimmingTherefore total time of biathlont(x)  =  ( 100 - x )/5    +  √[(50)² + x² ] / 2 Taking derivatives both sides of the equation we gett´(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 + x²t´(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 *xsquared         (50)²  +  x²   = 25/4 x²              2500 - 21/4 x²  =  0                            x²  =  2500*4/21  x  =  21,8 mTherefore she has to run  100 - 21,82  = 78,18 mAnd swim    √(50)² + (78,18)²   =  92m