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Parametric equations motion problem

  1. Apr 7, 2012 #1
    The question states:
    Two towns A and B are located directly opposite each other on a river 8km wide which flows at a speed 4km/h. A person from town A wants to travel to a town C located 6km up-stream from and on the same side as B. The person travels in a boat with maximum speed 10km/h and wishes to reach C in the shortest possible time. Let x(t) be the distance travelled upstream and y(t) be the distance travelled across the river in t hours. The person heads out at angle theta.

    a) Show that x(t)=10tcos(theta)-4t and y(t)=10tsin(theta)
    b) What is the angle theta and how long would the trip take?

    Relevant equations:
    So far I have used v=d/t along with some vector diagrams.

    My attempt:
    I have proven a) already by using v=d/t. The net velocity for x was equal to 10cos(theta)-4 and I just rearranged for x. I did the same to find y.

    I then found the angle theta by saying that sin(theta)=8/10, therefore theta=arcsin(4/5). Also, I found the theta in terms of arccos which was theta=arccos(3/5). I found these by using a distance triangle with adjacent=6, opposite=8 and hypotenuse=10.

    I then equated x(t)=6 ==> 10tcos(theta)-4t=6
    10tcos(arccos(3/5))-4t=6
    10t(3/5)-4t=6
    6t-4t=6
    t=3
    And equated y(t)=8 ==> 10tsin(theta)=8
    10tsin(arcsin(4/5))=8
    10t(4/5)=8
    8t=8
    t=1
    This is where I'm having problems. Shouldn't the time value be equal? If anyone could please help me out I would greatly appreciate it.
     
  2. jcsd
  3. Apr 7, 2012 #2

    tiny-tim

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