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Math Help - optimization problem

  1. #1
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    optimization problem

    A north-south highway intersects an east-west highway at point P. A vehicle crosses P at 1:00 p.m., travelling east at a onstant speed of 60 km/h. At the same instant, another vehicle is 5 km north of P, travelling south at 80 km/h. Find the time when the two vehicles are closest to each other and the distance between them at this time.
    Any help would be greatly appreciated!
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  2. #2
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    s_1=v_1t\text{ and } s_2=s_0-v_2t where v_1=60\text{ km/h}\quad v_2=80\text{ km/h}\quad s_0=5\text{ km}. The distance between them: d^2=s_1^2+s_2^2=v_1^2t^2+s_0^2-2s_0v_2t+v_2^2t^2=10000t^2-800t+25=f(t)
    f'(t)=20000t-800=0\Rightarrow t=\frac 1{25}\text{ h} so the time is 1:02:24, d=\sqrt{10000/625-800/25+25}=\mathbf{3\text{ km}}.
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