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Find a relation between dx/dt and dy/dt

  1. Mar 7, 2013 #1
    1. The problem statement, all variables and given/known data
    A particle moves counterclockwise around the ellipse with equation 9x^2 + 16y^2 = 25.

    a). In which of the four quadrants in dx/dt > 0? Explain.
    b). Find a relation between dx/dt and dy/dt.
    c). At what rate is the x-coordinate changing when the particle passes the point (-1,1) if its y-coordinate is increasing at a rate of 6 m/s?
    d). Find dy/dt when the particle is at the top and bottom of the ellipse.


    2. Relevant equations
    None


    3. The attempt at a solution

    a). I don't see how I could solve this with differentiation so I drew a picture of the ellipse. If the particle is travelling counter-clockwise, x will be increasing over time in quadrants 3 and 4.

    b). Implicitly differentiating for x and y both as functions of t I get
    dx/dt = (-32y*dy/dt)/18x

    c). Plugging in the values for the above formula...
    dx/dt = -32*6/18 = -32/3

    d). If the particle is at the top and bottom of the ellipse, then x is zero, and thus dy/dt is zero because

    dy/dt = (-18x*dx/dt)/32y = 0 @ x = 0


    Just checking if I did this correctly particularly a) as I don't see how I can "explicitly" show that dx/dt > 0 in quadrants 3 and 4 outside of explaining why in English.
     
  2. jcsd
  3. Mar 7, 2013 #2

    Mark44

    Staff: Mentor

    Since the ellipse was not given parametrically (i.e., as x = f(t) and y = g(t) for some functions f and g), I believe that you did what you were supposed to do for part a. IOW, look at the graph and determine visually that dx/dt > 0 where x is increasing.

    The only thing you should add are some units in part c. They're telling you the units at which y is changing, so you should report the same units when you say how x is changing. Also, it's probably a good idea to simplify the fraction.
     
    Last edited: Mar 7, 2013
  4. Mar 7, 2013 #3
    Will do, thanks.
     
    Last edited by a moderator: Mar 7, 2013
  5. Mar 7, 2013 #4

    Mark44

    Staff: Mentor

    What you can say for part a, is by direct observation, x is increasing in the 3rd and 4th quadrants.
     
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