Challenging related rate of a particle derivative problem?

Question

A particle moves counterclockwise around the ellipse 9x^2+16y^2=25.


a.In which of the four quadrants is the derivative dx/dt positive?
b.Find the 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 ft/s?
d. What is dy/dt when the particle is at the top and bottom of the ellipse?

I would really appreciate some help with this. Thank You.

Answer 1

a) in the third and fourth quadrants, since your x co-ordinate will be increasing as you move from third to fourth quadrants, so x' is positive.

b) 18 x x' + 32 y y' = 0

......x' = -32yy'/ 18x

c) x' = -32(6) / 18 = -32/ 3 ft/ sec { as you can see, x' is negative at (1, 1) a point in the 1st quadrant showing that x'<0 in the first quadrant}

d) at those points, x = 0, hence y' = -18x x' / 32y = 0

Answer 2

1) In upper half-plane: quadrants I and II.
2) 18x(dx/dt) + 32y(dy/dt) = 0
3) dx/dt = - 16/9 y/x (dy/dt) = -16/9*1/1*6 = - 32/3
4) dy/dt = 0

Answer 3

a. Since t always increases, we look for quadrants in which x is always increasing.