5. Consider a particle moving counter-clockwise around the ellipse
x^9/9 +y^2/4 = 1. Find a relationship
between
dy/dt
and
dx/dt
What are the rates of change of y and x at the times when the other variable isn't .
changing at all?
x^2/9 + y^2/4 = 1
or
4x^2 + 9y^2 = 36
8x dx/dt + 18y dy/dt = 0
so
dy/dt = - 8x dx/dt / 18y
and dy\x/dt : dx/dt = 4x : -9y
Just think about your last part of the question.
Suppose the x is not changing at all, which would mean of course that dx/dt = 0
what do you think would happen to y ?
if dx/dt = 0 then y should also be zero
@ reiny if dx/dt = 0 then y should also be zero
To find the relationship between dy/dt and dx/dt for the given ellipse, we can differentiate both sides of the equation with respect to time (t) using the chain rule.
Given the equation of the ellipse: (x^2/9) + (y^2/4) = 1
Differentiating both sides with respect to t:
d/dt [(x^2/9) + (y^2/4)] = d/dt (1)
Using the chain rule, we can differentiate each term separately:
(2x/9) * dx/dt + (2y/4) * dy/dt = 0
Simplifying the equation:
(2x/9) * dx/dt + (y/2) * dy/dt = 0
Now, we can rearrange this equation to express the relationship between dy/dt and dx/dt:
(dy/dt) / (dx/dt) = - (2x/9) / (y/2)
Cross multiplying:
2 * (dy/dt) = - (2x/9) * (dx/dt)
Simplifying further:
dy/dt = - (x/9) * (dx/dt)
Now, let's analyze the rates of change of y and x at the times when the other variable isn't changing at all.
When y is not changing (dy/dt = 0), we can substitute this value into the equation:
0 = - (x/9) * (dx/dt)
This equation implies that when y is not changing at all, dx/dt must also be zero. In other words, the rate of change of x is zero when the rate of change of y is zero.
Similarly, when x is not changing (dx/dt = 0), we can substitute this value into the equation:
dy/dt = - (x/9) * 0
This equation implies that when x is not changing at all, dy/dt will also be zero. In other words, the rate of change of y is zero when the rate of change of x is zero.
Therefore, the rates of change of y and x at the times when the other variable isn't changing at all are both zero.