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Optomization problem using integrals

  1. Apr 23, 2009 #1
    1. The problem statement, all variables and given/known data

    A boat leaves a dock at 2:00 pm and travels due south at a speed of 20 km/h. Another
    boat has been heading due east at 15 km/h and reaches the same dock at 3:00 pm. At
    what time were the two boats closest together?


    3. The attempt at a solution[/b

    We actually don't need to use integrals for this(and we haven't really learned how), but I want to.

    So a couple of quick questions...
    first, integrating 20 equals 20t+c correct?

    and second, either the 15km/h or the 20km/h should be negative, and it doesn't matter which one, right?

    If those two things are true, I should be all set, if not, I might have more questions. :p

    Thanks a lot for the help guys.
     
  2. jcsd
  3. Apr 23, 2009 #2

    Mark44

    Staff: Mentor

    You're overthinking this problem. Both boats are travelling at constant rates, so the distance they travel in t hours is going to be 15t km and 20t km, using the formula d = rt. That's all your integration has bought you, plus you have two constants of integration to worry about.

    If you haven't drawn a picture, you should, and maybe two of them, one for the positions of the two boats at 2:00 and another for their positions at 3:00. You need an expression that represents the distance between the two boats, as a function of t, and that's what you have to minimize, using differentiation.
     
  4. Apr 23, 2009 #3
    right, I have that position with the Pythagorean theorem. I have a^2+b^2=c^2. is a the position function of one boat and b the position function of the other boat? If that's the case, then I have c as a function of time if I integrate the velocities.

    The integration constants aren't a problem, because I know the position of one boat at t=0 and the other at t=1 so I can solve for c pretty easily.

    Then all I have to do is find the minimum of the Pythagorean function I created which looked to be 9/25.

    I'd add more of the math I've done to make what I'm saying more clear, but with latex down... :/
     
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