<p>A swimmer is 500 feet from the shore. She is traveling to a building 1800 feet down the shore. If she walks at 6 ft/sec and swims at 4ft/sec, how far from the building should she start walking so that she gets there in the shortest possible time?</p>
<p>You want to minimize TIME in this case.</p>
<p>TIME = WalkTime + SwimTime</p>
<p>using r = d/t, t= d/r.</p>
<p>So Walk Time is DistWalked/6 and SwimTime is DistSwim/4.</p>
<p>Draw a picture and let X be the distance down shore they should swim towards. Note that DistWalked is 1800-X and DistSwim (by Pyth Thm) is Sqrt(500^2+x^2).</p>
<p>Therefore TIME = (1800-X)/6 + Srt(500^2+x^2)/4</p>
<p>We want to minimize TIME, so we take a derivative and set it equal to ZERO. </p>
<p>D(Time) = -(1/6) +(1/8)<em>(500^2+x^2)^(-1/2)</em>(2X) = 0</p>
<p>(1/6) = (X/(4Sqrt(500^2+x^2)))</p>
<p>6x = 4Sqrt(500^2+x^2)</p>
<p>36x^2 = 16(500^2+x^2)</p>
<p>20x^2 = 16*500^2</p>
<p>x^2 = (4*500^2)/5</p>
<p>x = (2*500)/sqrt(5) = 1000sqrt(5)/5. This is how far down the beach they should aim...approximately 447 feet or yards down (whatever you used.)</p>